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THIS work was originally meant to be a continuation of the 
series "Electromagnetic Induction and its Propagation," 
published in The Electrician in 1885-6-7, but left unfinished. 
Owing, however, to the necessity of much introductory 
repetition, this plan was at once found to be impracticable, 
and was, by request, greatly modified. The result is some- 
thing approaching a connected treatise on electrical theory, 
though without the strict formality usually associated with 
a treatise. As critics cannot always find time to read more 
than the preface, the following remarks may serve to direct 
their attention to some of the leading points in this volume. 

The first chapter will, I believe, be found easy to read, 
and may perhaps be useful to many men who are accustomed 
to show that they are practical by exhibiting their ignorance 
of the real meaning of scientific and mathematical methods 
of enquiry. 

The second chapter, pp. 20 to 131, consists of an outline 
scheme of the fundamentals of electromagnetic theory from 
the Faraday-Maxwell point of view, with some small modifi- 
cations and extensions upon Maxwell's equations. It is done 
in terms of my rational units, which furnish the only way ot 
carrying out the idea of lines and tubes of force in a con- 
sistent and intelligible manner. It is also done mainly in 
terms of vectors, for the sufficient reason that vectors are 
the main subject of investigation. It is also done in the 
duplex form I introduced in 1885, whereby the electric and 



magnetic sides of electromagnetism are symmetrically ex- 
hibited and connected, whilst the "forces" and "fluxes" 
are the objects of immediate attention, instead of the 
potential functions which are such powerful aids to obscuring 
and complicating the subject, and hiding from view useful 
and sometimes important relations. 

The third chapter, pp. 132 to 305, is devoted to vector 
algebra and analysis, in the form used by me in my former 
papers. As I have at the beginning and end of this chapter 
stated my views concerning the unsuitability of quaternions 
for physical requirements, and my preference for a vector 
algebra which is based upon the vector and is dominated by 
vectorial ideas instead of quaternionic, it is needless to say 
more on the point here. But I must add that it has been 
gratifying to discover among mathematical physicists a con- 
siderable and rapidly growing appreciation of vector algebra 
on these lines; and moreover, that students who had found 
quaternions quite hopeless could understand my vectors very 
well. Regarded as a treatise on vectorial algebra, this chap- 
ter has manifest shortcomings. It is only the first rudiments 
of the subject. Nevertheless, as the reader may see from the 
applications made, it is fully sufficient for ordinary use in 
the mathematical sciences where the Cartesian mathematics 
is usually employed, and we need not trouble about more 
advanced developments before the elements are taken up. 
Now, there are no treatises on vector algebra in existence yet, 
suitable for mathematical physics, and in harmony with the 
Cartesian mathematics (a matter to which I attach the 
greatest importance). I believe, therefore, that this chapter 
may be useful as a stopgap. 

The fourth chapter, pp. 306 to 466, is devoted to the 
theory of plane electromagnetic waves, and, being mainly 
descriptive, may perhaps be read with profit by many who 
are unable to tackle the mathematical theory comprehen- 
sively. It may be also useful to have results of mathematical 


reasoning expanded into ordinary language for the benefit of 
mathematicians themselves, who are sometimes too apt to 
work out results without a sufficient statement of their 
meaning and effect. But it is only introductory to plane 
waves. Some examples in illustration thereof have been 
crowded out, and will probably be given in the next volume. 
I have, however, included in the present volume the applica- 
tion of the theory (in duplex form) to straight wires, and 
also an account of the effects of self-induction and leakage, 
which are of some significance in present practice as well 
as in possible future developments. There have been some 
very queer views promulgated officially in this country con- 
cerning the speed of the current, the impotence of self- 
induction, and other material points concerned. No matter 
how eminent they may be in their departments, officials need 
not be scientific men. It is not expected of them. But 
should they profess to be, and lay down the law outside their 
knowledge, and obstruct the spreading of views they cannot 
understand, their official weight imparts a fictitious impor- 
tance to their views, and acts most deleteriously in propagating 
error, especially when their official position is held up as a 
screen to protect them from criticism. But in other countries 
there is, I find, considerable agreement with my views. 

Having thus gone briefly through the book, it is desirable 
to say a few words regarding the outline sketch of electro- 
magnetics in the second chapter. Two diverse opinions have 
been expressed about it. On the one hand, it has been said 
to be too complicated. This probably came from a simple- 
minded man. On the other hand, it has been said to be too 
simple. This objection, coming from a wise man, is of 
weight, and demands some notice. 

Whether a theory can be rightly described as too simple 
depends materially upon what it professes to be. The pheno- 
mena involving electromagnetism may be roughly divided 
into two classes, primary and secondary. Besides the main 



primary phenomena, there is a large number of secondary 
ones, partly or even mainly electromagnetic, but also trenching 
upon other physical sciences. Now the question arises whether 
it is either practicable or useful to attempt to construct a 
theory of such comprehensiveness as to include the secondary 
phenomena, and to call it the theory of electromagnetism. I 
think not, at least at present. It might perhaps be done ii 
the secondary phenomena were thoroughly known ; but their 
theory is so much more debatable than that of the primary 
phenomena that it would be an injustice to the latter to too 
closely amalgamate them. Then again, the expression of the 
theory would be so unwieldy as to be practically useless ; the 
major phenomena would be apparently swamped by the minor. 
It would, therefore, seem best not to attempt too much, but 
to have a sort of abstract electromagnetic scheme for the 
primary phenomena only, and have subsidiary extensions 
thereof for the secondary. The theory of electromagnetism 
is then a primary theory, a skeleton framework corresponding 
to a possible state of things simpler than the real in innu- 
merable details, but suitable for the primary effects, and 
furnishing a guide to special extensions. From this point of 
view, the theory cannot be expressed too simply, provided it 
be a consistent scheme, and be sufficiently comprehensive to 
serve for a framework. I believe the form of theory in the 
second chapter will answer the purpose. It is especially 
useful in the duplex way of exhibiting the relations, which is 
clarifying in complicated cases as well as in simple ones. It 
is essentially Maxwell's theory, but there are some differences. 
Some are changes of form only ; for instance, the rationalisa- 
tion effected by changing the units, and the substitution ol 
the second circuital law for Maxwell's equation of electro- 
motive force involving the potentials, etc. But there is one 
change in particular which raises a fresh question. What is 
Maxwell's theory? or, What should we agree to understand 
by Maxwell's theory ? 

PREFACE. vii. 

The first approximation to the answer is to say, There is 
Maxwell's book as he wrote it ; there is his text, and there 
are his equations : together they make his theory. But when 
we come to examine it closely, we find that this answer is 
unsatisfactory. To begin with, it is sufficient to refer to 
papers by physicists, written say during the twelve years 
following the first publication of Maxwell's treatise, to see 
that there may be much difference of opinion as to what his 
theory is. It may be, and has been, differently interpreted by 
different men, which is a sign that it is not set forth in a per- 
fectly clear and unmistakeable form. There are many obscuri- 
ties and some inconsistencies. Speaking for myself, it was 
only by changing its form of presentation that I was able to 
see it clearly, and so as to avoid the inconsistencies. Now 
there is no finality in a growing science. It is, therefore, 
impossible to adhere strictly to Maxwell's theory as he gave it 
to the world, if only on account of its inconvenient form. 
But it is clearly not admissible to make arbitrary changes in 
it and still call it his. He might have repudiated them 
utterly. But if we have good reason to believe that the 
theory as stated in his treatise does require modification to 
make it self-consistent, and to believe that he would have 
admitted the necessity of the change when pointed out to him, 
then I think the resulting modified theory may well be called 

Now this state of things is exemplified by his celebrated 
circuital law defining the electric current in terms of magnetic 
force. For although he did not employ the other, or second 
circuital law, yet it may be readily derived from his equation 
of electromotive force ; and when this is done, and the law 
made a fundamental one, we readily see that the change it 
suffers in passing from the case of a stationary to that of a 
moving medium should be necessarily accompanied by a 
similar change in the first, or Maxwell's circuital law. An 
independent formal proof is unnecessary ; the similarity of 


form and of the conditions of motion show that Maxwell's 
auxiliary term in the electromotive force, viz., VqB (the 
motional electric force), where q is the velocity of the medium 
and B the induction, requires the use of a similar auxiliary 
term in the first circuital law, viz., VDq, the motional 
magnetic force, D being the displacement. And there is yet 
another change sometimes needed. For whilst B is circuital, 
so that a convective magnetic current does not appear in 
the second circuital equation, D is not always circuital, and 
convective electric current must therefore appear in the first 
circuital equation. For the reason just mentioned, it is the 
theory as thus modified that I consider to represent the true 
Maxwellian theory, with the other small changes required to 
make a fit. But further than this I should not like to go, 
because, having made a fit, it is not necessary, and because it 
would be taking too great a liberty to make additions without 
the strongest reason to consider them essential. 

The following example, which has been suggested to me 
by remarks in Prof. Lodge's recent paper on " Aberration 
Problems," referring to a previous investigation of Prof. J. J. 
Thomson, will illustrate the matter in question. It is known 
that if V be the speed of light through ether, the speed 
through a stationary transparent body, say water, is V//A, if p 
is the refractive index. Now what is the speed when the 
water is itself moving in the same direction as the light 
waves ? This is a very old problem. Fresnel considered that 
the external ether was stationary, and that the ether was /a 2 
times as dense in the water as outside, and that, when 
moving, the water only carried forward with it the extra ether 
it contained (or equivalently). This makes the speed of 
light referred to the external ether be V//* + v(l -ft~ 2 ), if v 
is the speed of the water. The experiments of Fizeau and 
Michelson have shown that this result is at least approxi- 
mately true, and there is other evidence to support FresnePs 
hypothesis, at least in a generalised form. But, in the case 


of water, the additional speed of light due to the motion of 
the water might be ^v instead of (1 - fir 2 ) v, without much 
disagreement. Now suppose we examine the matter electro- 
magnetically, and enquire what the increased speed through 
a moving dielectric should be. If we follow Maxwell's 
equations literally, we shall find that the extra speed is |r, 
provided i?/V is small. This actually seems to corroborate 
the experimental results. But the argument is entirely a 
deceptive one. Maxwell's theory is a theory of propagation 
through a simple medium. Fundamentally it is the ether, 
but when we pass to a solid or liquid dielectric it is still to be 
regarded as a simple medium in the same sense, because the 
only change occurring in the equations is in the value of one 
or both ethereal constants, the permittivity and inductivity 
practically only the first. Consequently, if we find, as above, 
that when the medium is itself moved, its velocity is not 
superimposed upon that of the velocity of waves through the 
medium at rest, the true inference is that there is something 
wrong with the theory. For all motion is relative, and it is 
an axiomatic truth that there should be superimposition of 
velocities, so that V//* + v should be the velocity in the above 
case according to any rational theory of propagation through 
a simple medium, the extra velocity being the full v t instead 
of Jv. And, as a matter of fact, if we employ the modified 
or corrected circuital law above referred to, we do obtain full 
superimposition of velocities. 

This example shows the importance of having a simply 
expressed and sound primary theory. For if the auxiliary 
hypotheses required to explain outstanding or secondary phe- 
nomena be conjoined to an imperfect primary theory we shall 
surely be led to wrong results. Whereas if the primary theory 
be good, there is at least a chance of its extension by auxiliary 
hypotheses being also good. The true conclusion from Fizeau 
and Michelson's results is that a transparent medium like 
water cannot be regarded as (in the electromagnetic theory) 


a simple medium like the ether, at least for waves of light, 
and that a secondary theory is necessary. Fresnel's sagacious 
speculation is justified, except indeed as regards its form of 
expression. The ether, for example, may be identical inside 
and outside the body, and the matter slip through it without 
sensibly affecting it. At any rate the evidence that this is the 
case preponderates, the latest being Prof. Lodge's experiments 
with whirling discs, though on the other hand must not be 
forgotten the contrary conclusion arrived at by Michelson as 
to the absence of relative motion between the earth and sur- 
rounding ether. But if the ether be stationary, Fresnel's 
speculation is roughly equivalent to supposing that the mole-' 
cules of transparent matter act like little condensers in increas- 
ing the permittivity, and that the matter, when in motion, 
only carries forward the increased permittivity. But however 
this matter may be finally interpreted, we must have a clear 
primary theory that can be trusted within its limits. Whether 
Maxwell's theory will last, as a sufficient and satisfactory 
primary theory upon which the numerous secondary deve- 
lopments required may be grafted, is a matter for the future 
to determine. Let it not be forgotten that Maxwell's theory 
is only the first step towards a full theory of the .ether ; and, 
moreover, that no theory of the ether can be complete that 
does not fully account for the omnipresent force of gravi- 

There is one other matter that demands notice in conclu- 
sion. It is not long since it was taken for granted that the 
common electrical units were correct. That curious and 
obtrusive constant 4?r was considered by some to be a sort of 
blessed dispensation, without which all electrical theory would 
fall to pieces. I believe that this view is now nearly extinct, 
and that it is well recognised that the 4?r was an unfortunate 
and mischievous mistake, the source of many evils. In plain 
English, the common system of electrical units involves an 
irrationality of the same kind as would be brought into the 


metric system of weights and measures, were we to define 
the unit area to be the area, not of a square with unit side, 
but of a circle of unit diameter. The constant TT would then 
obtrude itself into the area of a rectangle, and everywhere 
it should not be, and be a source of great confusion and 
inconvenience. So it is in the common electrical units, 
which are truly irrational. Now, to make a mistake is easy 
and natural to man. But that is not enough. The next 
thing is to correct it. When a mistake has once been started, 
it is not necessary to go on repeating it for ever and ever 
with cumulative inconvenience. 

The B. A. Committee on Electrical Standards had to do 
two kinds of work. There was the practical work of making 
standards from the experimentally found properties of matter 
(and ether). This has been done at great length, and with 
much labour and success. But there was also the theoretical 
work of fixing the relations of the units in a convenient, 
rational, and harmonious manner. This work has not yet 
been done. To say that they ought to do it is almost a 
platitude. Who else should do it ? To say that there is 
not at present sufficient popular demand for the change does 
not seem very satisfactory. Is it not for leaders to lead ? 
And who should lead but the men of light and leading who 
have practical influence in the matter ? 

Whilst, on the one hand, the immense benefit to be gained 
by rationalising the units requires some consideration to fully 
appreciate, it is, on the other hand, very easy to overestimate 
the difficulty of making the change. Some temporary incon- 
venience is necessary, of course. For a time there would be 
two sorts of ohms, &c., the old style and the new (or rational). 
But it is not a novelty to have two sorts of ohms. There 
have been several already. Eemember that the number of 
standards in present existence is as nothing to the number 
going to be made, and with ever increasing rapidity, by reason 
of the enormously rapid extension of electrical industries. 


Old style instruments would very soon be in a minority, and 
then disappear, like the pins. I do not know that there is a 
more important practical question than this one of rational- 
ising the units, on account of its far-reaching effect, and 
think that whilst the change could be made now with ease 
(with a will, of course), it will be far more troublesome if 
put off until the general British units are reformed; even 
though that period be not so distant as it is customary to 
believe. Electricians should set a good example. 

The reform which I advocate is somewhat similar to the 
important improvement made by chemists in their units 
about a quarter of a century ago. One day our respected 
master informed us that it had been found out that water 
was not HO, as he had taught us before, but something 
else. It was henceforward to be H 2 0. This was strange 
at first, and inconvenient, for so many other formulae had 
to be altered, and new books written. But no one questions 
the wisdom of the change. Now observe, here, that the 
chemists, when they found that their atomic weights were 
wrong, and their formulae irrational, did not cry " Too late," 
ignore the matter, and ask Parliament to legalise the old 
erroneous weights ! They went and set the matter right. 
Verb. sap. 

DECEMBER 16, 1893. 


[The dates within brackets are the dates of first publication.} 



1 7 [Jan. 2, 189 1.] Preliminary Remarks 1 

813 [Jan. 16, 1891.] On the nature of-Anti-Mathematicians and 

of Mathematical Methods of Enquiry ... ... 7 

14 16 [Jan. 30, 1891.] Description of some Electromagnetic Results 

deduced by Mathematical Reasoning ... 14 


(Pages 20 to 131.) 


20 [Feb. 13, 1891.] Electric and Magnetic Force ; Displacement and 

Induction ; Elastivity and Permittivity, Inductivity and Reluc- 
tivity 20 

21 Electric and Magnetic Energy ... ... ... 21 

22 Eolotropic Relations 22 

23 Distinction between Absolute and Relative Permittivity or 

Inductivity 23 

24 Dissipation of Energy. The Conduction-current ; Conductivity 

and Resistivity. The Electric Current ... ... ... ... 24 

25 Fictitious Magnetic Conduction - current and Real Magnetic 

Current 25 

26 [Feb. 27, 1891.] Forces and Fluxes 25 

27 Line-integral of a Force. Voltage and Gaussage 26 

28 Surface-integral of a Flux. Density and Intensity 27 

29 Conductance and Resistance ... ... ... ... ... ... 27 

30 Permittance and Elastance ... ... ... 28 

31 Permeance, Inductance and Reluctance ... 29 

32 Inductance of a Circuit 30 



33 [March 13, 1891.] Cross-connections of Electric and Magnetic 

Force. Circuital Flux. Circulation ... ... ... ... 32 

34 First Law of Circuitation ... ... 33 

35 Second Law of Circuitation ... ... ... ... 35 

36 Definition of Curl 35 

37 Impressed Force and Activity ... ... ... ... ... 36 

38 Distinction between Force of the Field and Force of the Flux ... 37 

39 [March 27, 1891.] Classification of Impressed Forces 38 

40 Voltaic Force 39 

41 Thermo-electric Force 40 

42 Intrinsic Electrisation 41 

43 Intrinsic Magnetisation ... ... ... ... ... ... 41 

44 The Motional Electric and Magnetic Forces. Definition of a 

Vector-Product 42 

45 Example. A Stationary Electromagnetic Sheet ... ... ... 43 

46 [April 17, 1891.] Connection between Motional Electric Force 

and " Electromagnetic Force " ... ... ... ... ... 44 

47 Variation of the Induction through a Moving Circuit ... ... 45 

48 Modification. Circuit fixed. Induction moving equivalently ... 46 

49 The Motional Magnetic Force 48 

50 The " Magneto-electric Force " ... 49 

51 Electrification and its Magnetic Analogue. Definition of " Diver- 

gence" ... ... ... ... ... ... ... ... 49 

52 A Moving Source equivalent to a Convection Current, and makes 

the True Current Circuital 51 

53 [May 1, 1891.] Examples to illustrate Motional Forces in a Moving 

Medium with a Moving Source. (1.) Source and Medium with 

a Common Motion. Flux travels with them undisturbed ... 53 

54 (2.) Source and Medium in Relative Motion. A Charge suddenly 

jerked into Motion at the Speed of Propagation. Generation 
of a Spherical Electromagnetic Sheet ; ultimately Plane. Equa- 
tions of a Pure Electromagnetic Wave ... ... ... ... 54 

55 (3.) Sudden Stoppage of Charge. Plane Sheet moves on. 

Spherical Sheet generated. Final Result, the Stationary 
Field 57 

56 (4.) Medium moved instead of Charge. Or both moved with 

same Relative Velocity ... ... ... ... ... ... 58 

57 (5.) Meeting of a Pair of Plane Sheets with Point-Sources. Can- 

celment of Charges ; or else Passage through one another ; 
different results. Spherical Sheet with two Plane Sheet 
Appendages ... ... ... ... ... ... ... ... 59 

58 (6.) Spherical Sheet without Plane Appendages produced by 

sudden jerking apart of opposite Charges ... ... ... 60 

59 (7.) Collision of Equal Charges of same Name 61 

60 (8.) Hemispherical Sheet. Plane, Conical, and Cylindrical Boun- 

daries 61 

61 General Nature of Electrified Spherical Electromagnetic Sheet ... 63 



62 [May 29, 1891.] General Remarks on the Circuital Laws. 
Ampere's Rule for deriving the Magnetic Force from the 

Current. Rational Current-element ... ... 64 

> 63 The Cardinal Feature of Maxwell's System. Advice to anti- 

Maxwellians . . . ... ... ... ... ... 66 

64 Changes in the Form of the First Circuital Law 67 

65 Introduction of the Second Circuital Law 68 

66 [July 3, 1891.'] Meaning of True Current. Criterion 70 

67 The Persistence of Energy. Continuity in Time and Space and 

Flux of Energy 72 

68 Examples. Convection of Energy and Flux of Energy due to an 

active Stress. Gravitational difficulty 74 

69 [July 17, 1891.] Specialised form of expression of the Con- 

tinuity of Energy ... ... ... ... ... ... ... 77 

70 Electromagnetic Application. Medium at Rest. The Poynting 

Flux 78 

71 Extension to a Moving Medium. Full interpretation of the 

Equation of Activity and derivation of the Flux of Energy ... 80 

72 Derivation of the Electromagnetic Stress from the Flux of 

Energy. Division into an Electric and a Magnetic Stress ... 83 

73 [July 31, 1891.] Uncertainty regarding the General Application 

of the Electromagnetic Stress ... ... ... ... ... 85 

74 The Electrostatic Stress in Air 87 

75 The Moving Force on Electrification, bodily and superficial. 

Harmonisation ... ... ... ... ... 90 

76 Depth of Electrified Layer on a Conductor ... ... ... 91 

77 [Aug. 21, 1891.] Electric Field disturbed by Foreign Body. 

Effect of a Spherical Non-conductor ... ... 93 

78 Dynamical Principle. Any Stress Self -equilibrating ... ... 95 

79 Electric Application of the Principle. Resultant Action on Solid 

Body independent of the Internal Stress, which is statically 
indeterminate. Real Surface Traction is the Stress Difference 96 

80 Translational Force due to Variation of Permittivity. Harmoni- 

sation with Surface Traction ... ... ... 98 

81 Movement of Insulators in Electric Field. Effect on the Stored 

Energy ... ... ... ... ... ... ... ... 99 

82 Magnetic Stress. Force due to Abrupt or Gradual Change of 

Inductivity. Movement of Elastically Magnetised Bodies ... 100 
82 A [Sept. 4, 1891.] Force on Electric Current Conductors. The 
Lateral Pressure becomes prominent, but no Stress Discon- 
tinuity in general ... ... ... ... ... 102 

83 Force on Intrinsically Magnetised Matter. Difficulty. Maxwell's 

Solution probably wrong. Special Estimation of Energy of a 
Magnet and the Moving Force it leads to ... 103 

84 Force on Intrinsically Electrised Matter 106 

85 Summary of the Forces. Extension to include varying States in 

a Moving Medium ... ... 107 



86 [Sept. 25, 1891.] Union of Electric and Magnetic to produce 

Electromagnetic Stress. Principal Axes ... ... ... 109 

87 Dependence of the Fluxes due to an Impressed Forcive upon 

its Curl only. General Demonstration of this Property ... 110 

88 Identity of the Disturbances due to Impressed Forcives having 

the same Curl. Example : A Single Circuital Source of 
Disturbance 112 

89 Production of Steady State due to Impressed Forcive by 

crossing of Electromagnetic Waves. Example of a Circular 
Source. Distinction between Source of Energy and of Dis- 
turbance ... ... ... ... ... ... ... ... 113 

90 [Oct. 16, 1891.] The Eruption of "47r"s 116 

91 The Origin and Spread of the Eruption 117 

92 The Cure of the Disease by Proper Measure of the Strength of 

Sources 119 

93 Obnoxious Effects of the Eruption 120 

94 A Plea for the Removal of the Eruption by the Radical Cure 122 

95 [Oct. 30, 1891.] Rational v. Irrational Electric Poles 123 

96 Rational v. Irrational Magnetic Poles 125 

APPENDIX A. [Jan. 23, 1891.] 



(Pages 132 to 305.) 


97 [Nov. 13, 1891.] Scalars and Vectors ... 132 

98 Characteristics of Cartesian and Vectdrial Analysis 133 

99 Abstrusity of Quaternions and Comparative Simplicity gained 

by ignoring them... ... ... ... ... ... ... 134 

100 Elementary Vector Analysis independent of the Quaternion . . . 136 

101 Tait v. Gibbs and Gibbs v. Tait 137 

102 Abolition of the Minus Sign of Quaternions 138 

103 [Dec. 4, 1891.] Type for Vectors. Greek, German, and Roman 

Letters unsuitable. Clarendon Type suitable. Typographical 
Backsliding in the Present Generation... ... ... ... 139 

L04 Notation. Tensor and Components of a Vector. Unit Vectors 

of Reference " 142 

L05 The Addition of Vectors. Circuital Property 143 

L06 Application to Physical Vectors. Futility of Popular Demon- 
strations. Barbarity of Euclid .. ... ... ... ... 147 

107 [Dec. 18, 1891.] The Scalar Product of Two Vectors. Notation 

and Illustrations ... ... ... ... ... 148 

108 Fundamental Property of Scalar Products, and Examples ... 151 

109 Reciprocal of a Vector ;. ... 155 



110 Expression of any Vector as the Sum of Three Indppendent 

Vectors 155 

111 {Jan. 1, 1892.] The Vector Product of Two Vectors. Illus- 

trations 156 

112 Combinations of Three Vectors. The Parallelepipedal Property 158 

113 Semi- Cartesian Expansion of a Vector Product, and Proof of 

the Fundamental Distributive Principle ... ... ... 159 

114 Examples relating to Vector Products ... ... ... ... 162 

115 [Jan. 29, 1892.} The Differentiation of Scalars and Vectors ... 163 

116 Semi -Cartesian Differentiation. Examples of Differentiating 

Functions of Vectors ... ... ... ... ... ... 165 

117 Motion along a Curve in Space. Tangency and Curvature ; 

Velocity and Acceleration ... ... . . 167 

118 Tortuosity of a Curve, and Various Forms of Expansion . . . 169 

119 [Feb. 12, 1892.} Hamilton's Finite Differentials Inconvenient 

and Unnecessary ... 172 

120 Determination of Possibility of Existence of Differential Co- 

efficients 174 

121 Variation of the Size and Ort of a Vector 176 

122 [March 4, 1892.} Preliminary on V. Axial Differentiation. 

Differentiation referred to Moving Matter ... 178 

123 Motion of a Rigid Body. Resolution of a Spin into other Spins 180 

124 Motion of Systems of Displacement, &c. ... ... ... ... 183 

125 Motion of a Strain-Figure 185 

126 [March 25, 1892.} Space-Variation or Slope VP of a Scalar 

Function ... ... ... ... ... ... ... ... 186 

127 Scalar Product VD. The Theorem of Divergence 188 

123 Extension of the Theorem of Divergence ... ... ... 190 

129 Vector Product WE, or the Curl of a Vector. The Theorem 

of Version, and its Extension ... ... ... ... ... 191 

130 [April 8, 1892.} Five Examples of the Operation of V in 

Transforming from Surface to Volume Summations 194 

131 Five Examples of the Operation of V in Transforming from 

Circuital to Surface Summations ... ... ... ... 197 

132 Nine Examples of the Differentiating Effects of V 199 

133 [May 13, 1892.} The Potential of a Scalar or Vector. The 

Characteristic Equation of a Potential, and its Solution ... 202 

1 34 Connections of Potential, Curl, Divergence, and Slope. Separa- 

tion of a Vector into Circuital and Divergent Parts. A Series 

of Circuital Vectors 206 

135 [May 27, 1892.} A Series of Divergent Vectors 209 

136 The Operation inverse to Divergence ... ... 212 

137 The Operation inverse to Slope 213 

138 The Operation inverse to Curl 214 

139 Remarks on the Inverse Operations ... ... ... ... 215 

140 Integration " by parts." Energy Equivalences in the Circuital 

Series , 216 



141 [June 10, 1892.} Energy and other Equivalences in the Diver- 

gent Series 217 

142 The Isotropic Elastic Solid. Relation of Displacement to Force 

through the Potential ... ._ 219 

143 The Stored Energy and the Stress in the Elastic Solid. The 

Forceless and Torqueless Stress ... ... ... ... 221 

144 Other Forms for the Displacement in terms of the Applied 

Forcive ... ... ... ... ... ... ... ... 224 

145 [June 24, 1892.} The Elastic Solid generalised to include 

Elastic, Dissipative and Inertial Resistance to Translation, 
Rotation, Expansion, and Distortion ... ... 226 

146 Electromagnetic and Elastic Solid Comparisons. First Ex- 

ample : Magnetic Force compared with Velocity in an In- 
compressible Solid with Distortional Elasticity ... ... 232 

147 Second Example : Same as last, but Electric Force compared 

with Velocity 234 

148 [July 15, 1892.} Third Example: A Conducting Dielectric 

compared with a Viscous Solid. Failure 234 

149 Fourth Example : A Pure Conductor compared with a Viscous 

Liquid. Useful Analogy ... ... ... ... ... 236 

150 Fifth Example : Modification of the Second and Fourth ... 239 

151 Sixth Example : A Conducting Dielectric compared with an 

Elastic Solid with Translational Friction 240 

152 Seventh Example : Improvement of the Sixth ... ... ... 21Q 

153 Eighth Example : A Dielectric with Duplex Conductivity com- 

pared with an Elastic Solid with Translational Elasticity and 
Friction. The singular Distortionless Case ... ... .. 241 

153A [July 29, 1892.} The Rotational Ether, Compressible or In- 
compressible ... ... ... ... ... ... ... 243 

154 First Rotational Analogy : Magnetic Force compared with 

Velocity 245 

155 Circuital Indeterminateness of the Flux of Energy in general 247 

156 Second Rotational Analogy : Induction compared with 

Velocity 249 

157 [Aug. 5, 1892.} Probability of the Kinetic Nature of Magnetic 

Energy 250 

158 Unintelligibility of the Rotational Analogue for a Conducting 

Dielectric when Magnetic Energy is Kinetic ... ... ... 252 

159 The Rotational Analogy, with Electric Energy Kinetic, ex- 

tended to a Conducting Dielectric by means of Translational 
Friction 253 

160 [Sept. 2, 1892.} Symmetrical Linear Operators, direct and in- 

verse, referred to the Principal Axes ... ... ... ... 256 

161 Geometrical Illustrations. The Sphere and Ellipsoid. Inverse 

Perpendiculars and Maccullagh's Theorem ... ... ... 259 

162 Internal Structure of Linear Operators. Manipulation of 

several when Principal Axes are Parallel ... ... ... 262 



163 [Sept. 16, 1802.] Theory of Displacement in an Eolotropic 

Dielectric. The Solution for a Point-Source... ... ... 264 

164 Theory of the Relative Motion of Electrification and the 

Medium. The Solution for a Point-Source in steady Recti- 
linear Motion. The Equilibrium Surfaces in general . . . 269 

165 [Sept. 30, 1892.'] Theory of the Relative Motion of Mag- 

netification and the Medium ... ... ... ... ... 274 

166 Theory of the Relative Motion of Magnetisation and the 

Medium. Increased Induction as well as Eolotropic Dis- 
turbance 277 

167 [Oct. 21, 1892.] Theory of the Relative Motion of Electric 

Currents and the Medium ... ... ... 281 

168 The General Linear Operator 283 

169 The Dyadical Structure of Linear Operators ... 285 

170 Hamilton's Theorem ... 287 

171 [Nov. 18, 1892.] Hamilton's Cubic and the Invariants con- 

cerned 289 

172 The Inversion of Linear Operators ... ... ... ... 293 

173 Vector Product of a Vector and a Dyadic. The Differentiation 

of Linear Operators ... ... ... ... ... ... 295 

174 [Dec. 9, 1892.] Summary of Method of Vector Analysis ... 297 

175 Uosuitability of Quaternions for Physical Needs. Axiom : 

Once a Vector, always a Vector ... ... ... ... 301 


(Pages 306 to 466.) 


176 [Dec. 30, 1892.] Action at a Distance versus Intermediate 

Agency. Contrast of New with Old Views about Electricity 306 

177 General Notions about Electromagnetic Waves. Generation of 

Spherical Waves and Steady States ... ... ... ... 310 

178 [Jan. 6, 1893.] Intermittent Source producing Steady States 

and Electromagnetic Sheets. A Train of S.H. Waves ... 314 

179 Self-contained Forced Electromagnetic Vibrations. Contrast 

with Static Problem 316 

180 Relations between E and H in a Pure Wave. Effect of Self- 

Induction. Fatuity of Mr. Preece's " KR law " 320 

181 [Jan. 27, 1893.] Wave-Fronts ; their Initiation and Progress 321 

182 Effect of a Non-Conducting Obstacle on Waves. Also of a 

Heterogeneous Medium ... ... . . ... ... ... 323 

183 Effect of Eolotropy. Optical Wave- Surf aces. Electromagnetic 

versus Elastic Solid Theories ... . . ... ... ... 325 

184 A Perfect Conductor is a Perfect Obstructor, but does not 

absorb the Energy of Electromagnetic Waves ... ... 328 



185 [Feb. 24, 1893.] Conductors at Low Temperatures 330 

186 Equilibrium of Radiation. The Mean Flux of Energy ... 331 

187 The Mean Pressure of Radiation 334 

188 Emissivity and Temperature .,. ... 335 

189 [March 10, 1893.] Internal Obstruction and Superficial Con- 

duction 337 

190 The Effect of a Perfect Conductor on External Disturbances. 

Reflection and Conduction of Waves ... ... 340 

L91 [March 24, 1893.] The Effect of Conducting Matter in 

Diverting External Induction ... .. ... ... ... 344 

192 Parenthetical Remarks on Induction, Magnetisation, Induc- 

tivity and Susceptibility 349 

193 [April 7, 1893.] Effect of a Thin Plane Conducting Sheet on 

a Wave. Persistence of Induction and Loss of Displacement 353 

194 The Persistence of Induction in Plane Strata, and in general. 

Also in Cores and in Linear Circuits .. ... ... ... 357 

195 [April 21, 1893.] The Laws of Attenuation of Total Displace- 

ment and Total Induction by Electric and Magnetic Conduc- 
tance 360 

196 The Laws of Attenuation at the Front of a Wave, due to 

Electric and Magnetic Conductance ... ... ... ... 564 

197 The Simple Propagation of Waves in a Distortionless Con- 

ducting Medium ... ... .. ... ... ... ... 366 

198 [May 5, 1893.] The Transformation by Conductance of an 

Elastic Wave to a Wave of Diffusion. Generation of Tails. 
Distinct Effects of Electric and Magnetic Conductance ... 369 

199 Application to Waves along Straight Wires 374 

200 [May 26, 1893.] Transformation of Variables from Electric 

and Magnetic Force to Voltage and Gaussage ... ... 378 

201 Transformation of the Circuital Equations to the Forms in- 

volving Voltage and Gaussage ... ... ... ... ... 381 

202 [June 9, 1893.] The Second Circuital Equation for Wires in 

Terms of V and C when Penetration is Instantaneous ... 386 

203 The Second Circuital Equation when Penetration is Not In- 

stantaneous. Resistance Operators, and their Definite 
Meaning ... ... ... ... ... ... ... ... 390 

204 Simply Periodic Waves Easily Treated in Case of Imperfect 

Penetration ... ... ... ... ... ... ... 393 

205 [July 7, 1893.} Long Waves and Short Waves. Identity of 

Speed of Free and Guided Waves ' ... 395 

206 The Guidance of Waves. Usually Two Guides. One sufficient, 

though with Loss. Possibility of Guidance within a Single 
Tube 399 

207 Interpretation of Intermediate or Terminal Conditions in the 

Exact Theory 401 

208 [Aug. 25, 1893.] The Spreading of Charge and Current in a 

long Circuit, and their Attenuation 403 



209 The Distortionless Circuit. No limiting Distance get by it 

when the Attenuation is ignored ... ... ... ... 409 

210 [Sept. 15, 1893.} The two Extreme Kinds of Diffusion in one 

Theory 411 

211 The Effect of varying the Four Line- Constanta as regards Dis- 

tortion and Attenuation ... ... ... 413 

212 The Beneficial Effect of Leakage in Submarine Cables 417 

213 [Oct. 6, 1893.} Short History of Leakage Effects on a Cable 

Circuit 420 

214 Explanation of Anomalous Effects. Artificial Leaks 424 

215 [Oct. 20. 1893.] Self-induction imparts Momentum to Waves. 

and that carries them on. Analogy with a Flexible Cord ... 429 

216 Self-induction combined with Leaks. The Bridge System of 

Mr. A. W. Heaviside, and suggested Distortionless Circuit ... 433 

217 [Nov. 3, 1893} Evidence in favour of Self-induction. Con- 

dition of First- Class Telephony. Importance of the Magnetic 
Eeactance 437 

218 Various ways, good and bad, of increasing the Inductance of 

Circuits 441 

219 [Nov. 17, 1893} Effective Resistance and Inductance of a 

Combination when regarded as a Coil, and Effective Con- 
ductance and Permittance when regarded as a Condenser ... 446 

220 Inductive Leaks applied to Submarine Cables 447 

221 General Theory of Transmission of Waves along a Circuit with 

or without Auxiliary Devices ... ... ... ... ... 449 

222 Application of above Theory to Inductive Leakance ... ... 453 


Parti. [July 14, 1893.} 455 

Part II. [Aug. 4,1893.} 463 




1. Preliminary Remarks. The main object of the series of 
articles of which this is the first, is to continue the work en- 
titled *' Electromagnetic Induction and its Propagation," com- 
menced in The Electrician on January 3, 1885, and continued to 
the 46th Section in September, 1887, when the great pressure 
on space and the want of readers appeared to necessitate its 
abrupt discontinuance. (A straggler, the 47th Section, appeared 
December 31, 1887.) Perhaps there were other reasons than 
those mentioned for the discontinuance. We do not dwell in 
the Palace of Truth. But, as was mentioned to me not long 
since, " There is a time coming when all things shall be found 
out." I am not so sanguine myself, believing that the well in 
which Truth is said to reside is really a bottomless pit. 

The particular branch of the subject which I was publishing 
in the summer of 1887 was the propagation of electromagnetic 
waves along wires through the dielectric surrounding them. 
This is itself a large and many-sided subject. Besides a 
general treatment, its many-sidedness demands that special 
cases of interest should receive separate full development. In 
general, the mathematics required is more or less of the charac- 
ter sometimes termed transcendental. This is a grandiloquent 
word, suggestive of something beyond human capacity to find 
out ; a word to frighten timid people into believing that it is 
all speculation, and therefore unsound. I do not know where 
transcendentality begins. You can find it in arithmetic. But 
never mind the word. What is of more importance is the fact 
that the interpretation of transcendental formulae is sometimes 


very laborious. Now the real object of true naturalists, in 
Sir W. Thomson's meaning of the word, when they employ 
mathematics to assist them, is not to make mathematical exer- 
cises (though that may be necessary), but to find out the con- 
nections of known phenomena, and by deductive reasoning, to 
obtain a knowledge of hitherto unknown phenomena. Any- 
thing, therefore, that aids this, possesses a value of its own wholly 
apart from immediate or, indeed, any application of the kind 
commonly termed practical. There is, however, practicality in 
theory as well as in practice. The very useful word " practi- 
cian " has lately come into use. It supplies a want, for it is 
evident the moment it is mentioned that a practician need not 
be a practical man ; and that, on the other hand, it may happen 
occasionally that a man who is not a practician may still be 
quite practical. 

2. Now, I was so fortunate as to discover, during the 
examination of a practical telephonic problem, that in a certain 
case of propagation along a conducting circuit through a con- 
ducting dielectric, the transcendentality of the mathematics 
automatically vanished, by the distorting effects on an electro- 
magnetic wave, of the resistance of the conductor, and of the 
conductance of the dielectric, being of opposite natures, so that 
they neutralised one another, and rendered the circuit non- 
distortional or distortionless. The mathematics was reduced, 
in the main, to simple algebra, and the manner of transmission 
of disturbances could be examined in complete detail in an 
elementary manner. Nor was this all. The distortionless 
circuit could be itself employed to enable us to understand the 
inner meaning of the transcendental cases of propagation, when 
the distortion caused by the resistance of the circuit makes the 
mathematics more difficult of interpretation. For instance, by 
a study of the distortionless circuit we are enabled to see not 
only that, but also why, self-induction is of such great import- 
ance in the transmission of rapidly-varying disturbances in 
preserving their individuality and preventing them from being 
attenuated to nearly nothing before getting from one end of a 
long circuit to the other ; and why copper wires are so success- 
ful in, and iron wires so prejudicial to effective, long-distance 
telephony. These matters were considered in Sections 40 to 


45 (June, July, August, 1887) of the work I have referred to, 
and Sections 46, 47 contain further developments. 

3. But that this matter of the distortionless circuit has, 
directly, important practical applications, is, from the purely 
scientific point of view, a mere accidental circumstance. Per- 
haps a more valuable property of the distortionless circuit is, 
that it is the Royal Road to electromagnetic waves in general, 
especially when the transmitting medium is a conductor as well 
as a dielectric. I have somewhat developed this matter in the 
Phil. Mag., 1888-9. Fault has been found with these articles 
that they are hard to read. They were harder, perhaps, to 
write. The necessity of condensation in a journal where space 
is so limited and so valuable, dealing with all branches of physi- 
cal science, is imperative. What is an investigator to do, when 
he can neither find acceptance of matter in a comparatively 
elementary form by journals of a partly scientific, partly tech- 
nical type, with many readers, nor, in a more learned form, by a 
purely scientific journal with comparatively few readers, and 
little space to spare ? To get published at all, he must con- 
dense greatly, and leave out all explanatory matter that he 
possibly can. Otherwise, he may be told his papers are more 
fit for publication in book form, and are therefore declined. 

There is a third course, of course, viz., to keep his investiga- 
tions to himself. But that does not answer, in a general way, 
though it may do so sometimes. It is like putting away seed 
in a mummy case, instead of planting it, and letting it take its 
chance of growing to a useful plant. There is nothing like 
publication and free criticism for utility. I can see only one 
good excuse for abstaining from publication when no obstacle 
presents itself. You may grow your plant yourself, nurse it 
carefully in a hot-house, and send it into the world full-grown. 
But it cannot often occur that it is worth the trouble taken. 
As for the secretiveness of a Cavendish, that is utterly inex- 
cusable ; it is a sin. It is possible to imagine the case of a man 
being silent, either from a want of confidence in himself, or 
from disappointment at the reception given to, and want of 
appreciation of, the work he gives to the world ; few men have 
an unbounded power of persistence ; but to make valuable dis- 
coveries, and to hoard them up as Cavendish did, without any 

B 2 


valid reason, seems one of the most criminal acts such a man 
could be guilty of. This seems strong language, but as Prof. Tait 
tells us that it is almost criminal not to know several foreign 
languages, which is a very venial offence in the opinion of others, 
it seems necessary to employ strong language when the crimi- 
nality is more evident. (See, on this point, the article in The 
Electrician, November 14, 1890. It is both severe and logical.) 

4. I had occasion, just lately, to use the word " naturalist." 
The matter involved here is worthy of parenthetical considera- 
tion. Sir W. Thomson does not like " physicist," nor, I think, 
"scientist" either. It must, however, be noted that the 
naturalist, as at present generally understood, is a student of 
living nature only. He has certainly no exclusive right to so 
excellent a name. On the other hand, the physicist is a 
student of inanimate nature, in the main, so that he has no 
exclusive right to the name, either. Both are naturalists. But 
their work is so different, and their type of mind also so 
different, that it seems very desirable that their names should 
be differentiated, and that " naturalist," comprehending both, 
should be subdivided. Could not one set of men be induced to call 
themselves organists? We have organic chemistry, and organisms, 
and organic science ; then why not organists 1 Perhaps, how- 
ever, organists might not care to be temporarily confounded 
with those members of society who earn their living by setting a 
cylinder in rotatory motion. If so, there is another good name, 
viz., vitalist, for the organist, which would not have any ludic- 
rous association. Then about the other set of men. Are they 
not essentially students of the properties of matter, and there- 
fore materialists? That "materialist" is the right name is 
obvious at a glance. Here, however, a certain suppositions 
evil association of the word might militate against its adoption. 
But this would be, I think, an unsound objection, for I do not 
think there is, or ever was, such a thing as a materialist, in 
the supposed evil sense. Let that notion go, and the valuable 
word "materialist" be put to its proper use, and be dignified 
by association with an honourable body of men. 

Buffon, Cuvier, Darwin, were typical vitalises. 
Newton, Faraday, Maxwell, were typical materialists. 


All were naturalists. For my part I always admired the old- 
fashioned term "natural philosopher." It was so dignified, and 
raised up visions of the portraits of Count Rumford, Young, 
Herschel, Sir H. Davy, &c., usually highly respectable-looking 
elderly gentlemen, with very large bald heads, and much 
wrapped up about the throats, sitting in their studies ponder- 
ing calmly over the secrets of nature revealed to them by their 
experiments. There are no natural philosophers now-a-days. 
How is it possible to be a natural philosopher when a Salvation 
Army band is performing outside ; joyously, it may be, but not 
most melodiously ? But I would not disparage their work ; it 
may be far more important than his. 

5. Returning to electromagnetic waves. Maxwell's in- 
imitable theory of dielectric displacement was for long gene- 
rally regarded as a speculation. There was, for many years, an 
almost complete dearth of interest in the unverified parts of 
Maxwell's theory. Prof. Fitzgerald, of Dublin, was the most 
prominent of the very few materialists (if I may use the word) 
who appeared to have a solid faith in the electromagnetic theory 
of the ether ; thinking about it and endeavouring to arrive at an 
idea of the nature of diverging electromagnetic waves, and how to 
produce them, and to calculate the loss of energy by radiation. 
An important step was then made by Poynting, establishing the 
formula for the flow of energy. Still, however, the theory 
wanted experimental proof. Three years ago electromagnetic 
waves were nowhere. Shortly after, they were everywhere. 
This was due to a very remarkable and unexpected event, 
no less than the experimental discovery by Hertz, of Karlsruhe 
(now of Bonn), of the veritable actuality of electromagnetic 
waves in the ether. And it never rains but it pours ; for whilst 
Hertz with his resonating circuit was working in Germany 
(where one would least expect such a discovery to be made, if one 
judged only by the old German electro-dynamic theories), Lodge 
was doing in some respects similar work in England, in connec- 
tion with the theory of lightning conductors. These researches, 
followed by the numerous others of Fitzgerald and Trouton, 
J. J. Thomson, &c., have dealt a death-blow to the electro- 
dynamic speculations of the Weber-Clausius type (to mention 
only the first and one of the last), and have given to Maxwell's 


theory just what was wanted in its higher parts, more experi- 
mental basis. The interest excited has been immense, and the 
theorist can now write about electromagnetic waves without in- 
curring the reproach that he is working out a mere paper theory. 
The speedy recognition of Dr. Hertz by the Royal Society is a 
very unusual testimony to the value of his researches. 

At the same time I may remark that to one who had care- 
fully examined the nature of Maxwell's theory, and looked into 
its consequences, and seen how rationally most of the pheno- 
mena of electromagnetism were explained by it, and how it 
furnished the only approximately satisfactory (paper) theory of 
light known ; to such a one Hertz's demonstration came as a 
matter of course only it came rather unexpectedly. 

6. It is not by any means to be concluded that Maxwell 
spells Finality. There is no finality. It cannot even be accu- 
rately said that the Hertzian waves prove Maxwell's dielectric 
theory completely. The observations were very rough indeed, 
when compared with the refined tests in other parts of electrical 
science. The important thing proved is that electromagnetic 
waves in the ether at least approximately in accordance with 
Maxwell's theory are a reality, and that the Faraday-Maxwellian 
method is the correct one. The other kind of electrodynamic 
speculation is played out completely. There will be plenty of 
room for more theoretical speculation, but it must now be of 
the Maxwellian type, to be really useful. 

7. In what is to follow, the consideration of electromagnetic 
waves will (perhaps) occupy a considerable space. How much 
depends entirely upon the reception given to the articles. 
Mathematics is at a discount, it seems. Nevertheless, as the 
subject is intrinsically a mathematical one, I shall not scruple 
to employ the appropriate methods when required. The reader 
whose scientific horizon is bounded entirely by commercial con- 
siderations may as well avoid these articles. Speaking without 
prejudice, matter more to his taste may perhaps be found under 
the heading TRADE NOTICES.* Sunt quos curricula. 

I shall, however, endeavour to avoid investigations of a com- 
plex character ; also, when the methods and terms used are not 

* Referring to The Electrician, in which this work first appeared. 


generally known I shall explain them. Considering the lapse 
of time since the discontinuance of E.M.I, and its P. it would 
be absurd to jump into the middle of the subject all at once. It 
. must, therefore, be gradually led up to. I shall, therefore, in 
the next place make a few remarks upon mathematical investi- 
gations in general, a subject upon which there are many popular 
delusions current, even amongst people who, one would think, 
should know better. 

8. There are men of a certain type of mind who are never 
wearied with gibing at mathematics, at mathematicians, and at 
mathematical methods of inquiry. It goes almost without say- 
ing that these men have themselves little mathematical bent. 
I believe this to be a general fact ; but, as a fact, it does not 
explain very well their attitude towards mathematicians. The 
reason seems to lie deeper. How does it come about, for in- 
stance, that whilst they are themselves so transparently ignorant 
of the real nature, meaning, and effects of mathematical investi- 
gation, they yet lay down the law in the most confident and 
self-satisfied manner, telling the mathematician what the nature 
of his work is (or rather is not), and of its erroneousness and 
inutility, and so forth 1 It is quite as if they knew all about it. 

It reminds one of the professional paradoxers, the men who 
want to make you believe that the ratio of the circumference 
to the diameter of a circle is 3, or 3*125, or some other nice 
easy number (any but the right one) ; or that the earth is flat, 
or that the sun is a lump of ice ; or that the distance of the 
moon is exactly 6 miles 500 yards, or that the speed of the 
current varies as the square of the length of the line. They, 
too, write as if they knew all about it ! Plainly, then, the 
anti-mathematician must belong to the same class as the 
paradoxer, whose characteristic is to be wise in his ignorance, 
whereas the really wise man is ignorant in his wisdom. But 
this matter may be left for students of mind to settle. What 
is of greater importance is that the anti-mathematicians some- 
times do a deal of mischief. For there are many of a neutral 
frame of mind, little acquainted themselves with mathematical 
methods, who are sufficiently impressible to be easily taken in 
by the gibers and to be prejudiced thereby ; and, should 
they possess some mathematical bent, they may be hindered 


by their prejudice from giving it fair development. We 
cannot all be Newtons or Laplaces, but that there is an 
immense amount of moderate mathematical talent lying latent 
in the average man I regard as a fact ; and even the moderate 
development implied in a working knowledge of simple alge- 
braical equations can, with common-sense to assist, be not 
only the means of valuable mental discipline, but even be of 
commercial importance (which goes a long way with some 
people), should one's occupation be a branch of engineering for 

9. " Mathematics is gibberish." Little need be said about 
this statement. It is only worthy of the utterly illiterate. 

" What is the use of it ? It is all waste of time. Better be 
doing something useful. Why, you might be inventing a new 
dynamo in the time you waste over all that stuff." Now, 
similar remarks to these I have often heard from fairly intelli- 
gent and educated people. They don't see the use of it, that is 
plain. That is nothing ; what is to the point is that they con- 
clude that it is of no use. For it may be easily observed that 
the parrot-cry "What's the use of it?" does not emanate in a 
humble spirit of inquiry, but on the contrary, quite the reverse. 
You can see the nose turn up. 

But what is the use of it, then ? Well, it is quite certain that 
if a person has no mathematical talent whatever he had really 
better be doing something " useful," that is to say, something 
else than mathematics, (inventing a dynamo, for instance,) and 
not be wasting his time in (so to speak) trying to force a crop 
of wheat on the sands of the sea-shore. This is quite a personal 
question. Every mind should receive fair development (in 
good directions) for what it is capable of doing fairly well. 
People who do not cultivate their minds have no conception of 
what they lose. They become mere eating and drinking and 
money-grabbing machines. And yet they seem happy ! There 
is some merciful dispensation at work, no doubt. 

" Mathematics is a mere machine. You can't get anything 
out of it that you don't put in first. You put it in, and then 
just grind it out again. You can't discover anything by 
mathematics, or invent anything. You can't get more than a 
pint out of a pint pot." And so forth. 


It is scarcely credible to the initiated that such statements 
could be made by any person who could be said to have an in- 
tellect. But I have heard similar remarks from really talented 
men, who might have fair mathematical aptitude themselves, 
though quite undeveloped. The fact is, the statements contain 
at once a profound truth, and a mischievous fallacy. That the 
fallacy is not self-evident affords an excuse for its not being 
perceived even by those who may (perhaps imperfectly) recog- 
nise the element of truth.. But as regards the truth men- 
tioned, I doubt whether the caviller has generally any distinct 
idea of it either, or he would not express it so contemptuously 
along with the fallacy. 

10. By any process of reasoning whatever (not fancy) you 
cannot get any results that are not implicitly contained in the 
material with which you work, the fundamental data and their 
connections, which form the basis of your inquiry. You may 
make mistakes, and so arrive at erroneous results from the 
most correct data. Or the data may be faulty, and lead to 
erroneous conclusions by the most correct reasoning. And in 
general, if the data be imperfect, or be only true within cer- 
tain limits hardly definable, the results can have but a limited 
application. Now all this obtains exactly in mathematical 
reasoning. It is in no way exempt from the perils of reasoning 
in general. But why the mathematical reasoning should be 
singled out for condemnation as mere machine work, dependent 
upon what the machine is made to do, with a given supply of 
material, is not very evident. The cause lies deep in the 
nature of the anti-mathematician ; he has not recognised that 
all reasoning must be, in a sense, mechanical, else it is not 
sound reasoning at all, but vitiated by fancy. 

Mathematical reasoning is, fundamentally, not different 
from reasoning in general. And as by the exercise of the 
reason discoveries can be made, why not by mathematical 
reasoning? Whatever were Newton and the long array of 
mathematical materialists who followed him doing all the time? 
Making discoveries, of course, largely assisted by their mathe- 
matics. I say nothing of the pure mathematicians. Their 
discoveries are extensions of the field of mathematics itself a 
perfectly limitless field. I refer only to students of Nature on 


its material side, who have employed mathematics expressly 
for the purpose of making discoveries. Some of the unmathe- 
matical believe that the mathematician is merely engaged in 
counting or in doing long sums ; this probably arises from 
reminiscences of their schooldays, when they were flogged 
over fractions. Now this is only a part of his work, a some- 
times necessary and very disagreeable part, which he would 
willingly hand over to a properly trained computator. This 
part of the work only concerns the size of the effects, but 
it is the effects themselves to which I refer when I speak of 

11. Mathematics is reasoning about quantities. Even if 
qualities are in question, it is their quantities that are subjected 
to the mathematics. If there be something which cannot be 
reduced to a quantity, or more generally to a definite function, 
no matter how complex and involved, of any number of other 
quantities which can be measured (either actually, or in 
imagination), then that something cannot be accurately reasoned 
about, because it is in part unknown. Not unknown in the sense 
in which a quantity is said to be unknown in algebra, when it 
is virtually known because virtually expressible in terms of 
known quantities, but literally unknown by the absence of 
sufficient quantitative connection with the known. Thus only 
the known can be accurately reasoned about. But this in- 
cludes, it will be observed, everything that can be deduced 
from the known, without appeal to the unknown. The un- 
known is not necessarily unknowable ; fresh knowns may make 
the former unknowns become also known. The distinction is 
a very important one. The limits of human knowledge are 
ever shifting. But there must be an ultimate limit, because 
we are a part of Nature, and cannot go beyond it. Beyond 
this limit, the Unknown becomes the Unknowable, which it is 
of little service to discuss, though it will always be a favourite 
subject of speculation. But whatever is in this Universe can 
be (or might be) found out, and therefore does not belong to 
the unknowable. Thus the constitution of the middle of the 
sun, or of the ether, or the ultimate nature of magnetisation, 
or of universal gravitation, or of life, are not unknowable ; and 
this statement is true, even though they should never be dis- 


covered. There are no inscrutables in Nature. By Faith only 
can we go beyond as far and where we please. 

Human nature, or say a man, is a highly complex quantity. 
We are compelled to take him in parts, and consider this or 
that quality, and imagine it measured and brought into proper 
connection with other qualities and external influences. Yet 
a man, if we only knew him intimately enough, could be 
formularised, and have his whole life-history developed. Even 
the universe itself, if every law in action were thoroughly 
known, could have its history, past, present, and tp come, 
formularised down to the minutest particulars, provided no 
discontinuity or special act of creation occur. But even the 
special act of creation could be formularised, and its effects 
deduced, if we knew in what it consisted. And special acts of 
creation might be going on continuously, involving continuous 
changes in the laws of nature, and could be formularised, if 
the acts of creation were known, or the so to speak law of 
the discontinuities. The case is somewhat analogous to that 
of impressed forces acting upon a dynamical system. The 
behaviour of the system is perfectly definite and formularisable 
so long as no impressed forces act, and ceases to be definite if 
unknown impressed forces act. But if the forces be also 
known, then the course of events is again definitely formularis- 
able. The assumption of a special act of creation, either now 
or at any time, is merely a confession of ignorance. We have 
no evidence of any such discontinuities. We cannot prove that 
there have never been any; nor can we prove that the sun will 
not rise to-morrow, or that the clock will not wind itself up 
again when the weight has run down. 

12. Nearly all the millions, or rather billions, of human 
beings who have peopled this earth have been content to go 
through life taking things as they found them, and without 
any desire to understand what is going on around them. It is 
exceedingly remarkable that the scientific spirit (asking how it 
is done), which is so active and widespread at the present day, 
should be of such recent origin. With a few exceptions, it 
hardly existed amongst the Ancients (who would be more appro- 
priately termed the Youngsters). It is a very encouraging fact 
for evolutionists, leading them to believe that the evolution of 


man is not played out ; but that man is capable, intellectually, 
of great development, and that the general standard will be 
far higher in the future than at present. 

Now, in the development of our knowledge of the workings 
of Nature out of the tremendously complex assemblage of 
phenomena presented to the scientific inquirer, mathematics 
plays in some respects a very limited, in others a very impor- 
tant part. As regards the limitations, it is merely necessary to 
refer to the sciences connected with living matter, and to the 
ologie^ generally, to see that the facts and their connections 
are too indistinctly known to render mathematical analysis 
practicable, to say nothing of the complexity. Facts are of 
not much use, considered as facts. They bewilder by their 
number and their apparent incoherency. Let them be digested 
into theory, however, and brought into mutual harmony, and 
it is another matter Theory is the essence of facts. Without 
theory scientific knowledge would be only worthy of the mad- 

In some branches of knowledge, the facts have been so far 
refined into theory that mathematical reasoning becomes ap- 
plicable on a most extensive scale. One of these branches is 
Electromagnetism, that most extensive science which presents 
such a remarkable two-sidedness, showing the electric and the 
magnetic aspects either separately or together, in stationary 
conditions, and a third condition when the electric and mag- 
netic forces act suitably in dynamical combination, with equal 
development of the electric and magnetic energies, the state of 
electromagnetic waves. 

It goes without saying that there are numerous phenomena 
connected with electricity and magnetism which are very 
imperfectly understood, and which have not been formularised, 
except perhaps in an empirical manner. Such is particularly 
the case where the sciences of Electricity and Chemistry meet. 
Chemistry is, so far, eminently unmathematical (and therefore 
a suitable study for men of large capacity, who may be nearly 
destitute of mathematical talent but this by the way), and it 
appears to communicate a part of its complexity and vagueness 
to electrical science whenever electrical phenomena which we 
can study are accompanied by chemical changes. But generally 
speaking, excepting electrolytic phenomena and other compli- 


cations (e.g., the transport of matter in rarefied media when 
electrical discharges occur), the phenomena of electromagnetisni 
are, in the main, remarkably well known, and amenable to 
mathematical treatment. 

13. Ohm (a distinguished mathematician, be it noted) 
brought into order a host of puzzling facts connecting electro- 
motive force and electric current in conductors, which all pre- 
vious electricians had only succeeded in loosely binding together 
qualitatively under some rather vague statements. Even as 
late as 20 years ago, "quantity" and "tension" were much used 
by men who did not fully appreciate Ohm's law. (Is it not 
rather remarkable that some of Germany's best men of genius 
should have been, perhaps, unfairly treated ? Ohm ; Mayer ; 
Reis ; even von Helmholtz has mentioned the difficulty he had 
in getting recognised. But perhaps it is the same all the 
world over.) Ohm found that the results could be summed 
up in such a simple law that he who runs may read it, and a 
schoolboy now can predict what a Faraday then could only 
guess at roughly. By Ohm's discovery a large part of the 
domain of electricity became annexed to theory. Another 
large part became virtually annexed by Coulomb's discovery of 
the law of inverse squares, and completely annexed by Green's 
investigations. Poisson attacked the difficult problem of in- 
duced magnetisation, and his results, though differently 
expressed, are still the theory, as a most important first 
approximation. Ampere brought a multitude of phenomena 
into theory by his investigations of the mechanical forces 
between conductors supporting currents and magnets. Then 
there were the remarkable researches of Faraday, the prince of 
experimentalists, on electrostatics and electrodynamics and the 
induction of currents. These were rather long in being brought 
from the crude experimental state to a compact system, ex- 
pressing the real essence. Unfortunately, in my opinion, 
Faraday was not a mathematician. It can scarcely be doubted 
that had he been one, he would have been greatly assisted in 
his researches, have saved himself much useless speculation, 
and would have anticipated much later work. He would, for 
instance, knowing Ampere's theory, by his own results have 
readily been led to Neumann's theory, and the connected 


work of Helmholtz and Thomson. But it is perhaps too much 
to expect a man to be both the prince of experimentalists and 
a competent mathematician. 

Passing over the other developments which were made in 
the theory of electricity and magnetism, without striking new 
departures, we come to about 1860. There was then a collec- 
tion of detached theories, but loosely connected, and embedded 
in a heap of unnecessary hypotheses, scientifically valueless, and 
entirely opposed to the spirit of Faraday's ways of thinking, 
and, in fact, to the spirit of the time. All the useless hypo- 
theses had to be discarded, for one thing ; a complete and 
harmonious theory had to be made up out of the useful re- 
mainder, for another; and, in particular, the physics of the 
subject required to be rationalised, the supposed mutual attrac- 
tions or repulsions of electricity, or of magnetism, or of elements 
of electric currents upon one another, abolished, and electro- 
magnetic effects accounted for by continuous actions through 
a medium, propagated in time. All this, and much more, was 
done. The crowning achievement was reserved for the heaven- 
sent Maxwell, a man \v ^e fame, great as it is now, has, com- 
paratively speaking, yet to come. 


14. It will have been observed that I have said next to 
nothing upon the study of pure, mathematics ; this is a matter 
with which we are not concerned. But that I have somewhat 
dilated (and I do not think needlessly) upon the advantages 
attending the use of mathematical methods by the materialist 
to assist him in his study of the laws governing the material 
universe, by the proper co-ordination of known and the dis- 
covery of unknown (but not unknowable) phenomena. 

It was discovered by mathematical reasoning that when an 
electric current is started in a wire, it begins entirely upon its 
skin, in fact upon the outside of its skin ; and that, in conse- 
quence, sufficiently rapidly impressed fluctuations of the current 
keep to the skin of the wire, and do not sensibly penetrate to 
its interior. 

Now very few (if any) unmathematical electricians can 
understand this fact ; many of them neither understand it nor 
believe it. Even many who do believe it do so, I believe, simply 
because they are told so, and not because they can in the least 


feel positive about its truth of their own knowledge. As an 
eminent practician remarked, after prolonged scepticism, " When 
Sir W. Thomson says so, who can doubt it ? " What a world of 
worldly wisdom lay in that remark ! 

Now I do admire this characteristically stubborn English way 
of being determined not to be imposed upon by any absurd 
theory that goes against all one's most cherished convictions, 
and which cannot be properly understood without mathe- 
matics. For without the mathematics, and with only the 
sure knowledge of Ohm's law and the old-fashioned notions 
concerning the function of a conducting wire to guide one, no 
one would think of such a theory. It is quite preposterous 
from this point of view. Nevertheless, it is true ; and the 
view was not put forward as a hypothesis, but as a plain matter 
of fact. 

The case in question is one in which we can be very sure of 
all the fundamental data of any importance, and the laws con- 
cerned. We can, for instance, by straightforward experiment, 
especially with properly constructed induction balances ad- 
mitting of exact interpretation of resu '; readily satisfy our- 
selves that a high degree of accuracy must obtain not merely 
for Ohm's law, but also for Jbo,xiday's law of E.M.F. in circuits, 
and even in iron for Poisson's law of induced magnetisation, 
within certain limits. We have, therefore, all the conditions 
wanting for the successful application of mathematical reasoning 
of a precise character, and justification for the confidence that 
mathematicians can feel in the results theoretically deduced in 
a legitimate manner, however difficult it may be to give an 
easily intelligible account of their meaning to the unmathe- 

This, however, I will say for the sceptic who has the courage 
of his convictions, and writes openly against what is, to him, 
pure nonsense. He is doing, in his way, good service in the 
cause of truth and the advancement of scientific knowledge, by 
stimulating interest in the subject and causing people to inquire 
and read and think about these things, and form their own 
judgment if possible, and modify their old views if they should 
be found wanting. Nothing is more useful than open and free 
criticism, and the truly earnest and disinterested student of 
science always welcomes it. 


15. The following may assist the unmathematical reader to 
an understanding of the subject. It is not demonstrative, of 
course, but is merely descriptive. If, however, it be translated 
into mathematical language and properly worked out, it will 
be found to be demonstrative, and to lead to a complete theory 
of the functions of wires in general. 

Start with a very long solenoid of fine wire in circuit with a 
source of electrical energy. Let the material inside the solenoid 
be merely air, that is to say, ether and air. If we examine the 
nature of the fluctuations of current in the coil in relation to 
the fluctuations of impressed force on it, we find that the cur- 
rent in the coil behaves as if it were a material fluid possessing 
inertia and moving against resistance. The fanatics of Ohm's 
law do not usually take into account the inertia. It is as if the 
current in the coil could not move without simultaneously 
setting into rotation a rigid material core filling the solenoid, 
and free to rotate on its axis. 

If we take the air out of the solenoid and substitute any other 
non-conducting material for a core the same thing happens; 
only the inertia varies with the material, according to its mag- 
netic inductivity. 

But if we use a conducting core we get. new phenomena, 
for we find that there is no longer a definite resistance and a 
definite inertia. There is now frictional resistance in the core, 
and this increases the effective resistance of the coil. At the 
same time the inertia is reduced. 

On examining the theory of the matter (on the basis of 
Ohm's law and Faraday's law applied to the conducting core) 
we find that we can now account for things (in our analogy) by 
supposing that the rigid solid core first used is replaced by a 
viscous fluid core, like treacle. On starting a current in the 
coil it cannot now turn the core round bodily at once, but only 
its external portion. In time, however, if the source of current 
be steadily operating, the motion will penetrate throughout the 
viscous core, which will finally move as the former rigid core did. 
If, however, the current in the coil fluctuate in strength very 
rapidly, the corresponding fluctuations of motion in the core 
will be practically confined to its skin. The effective inertia is 
reduced because the core does not move as a rigid body ; the 
effective resistance is increased by the viscosity generating heat.- 


Now, returning to the solenoid, we have perfect symmetry 
with respect to its axis, since the core is supposed to be exceed- 
ingly long, and uniformly lapped with wire. The situation of 
the source of energy, as regards the core itself, is plainly on its 
boundary, where the coil is placed ; and it is therefore a matter 
of common-sense that in the communication of energy to the 
core either when the current is steady or when it varies, the 
transfer of energy takes place transversely, that is, from the 
boundary to the axis, in planes perpendicular to the axis, and 
therefore perpendicular to the current in the core itself. This 
is confirmed by the electromagnetic equations. 

But the electromagnetic equations go further than this, and 
assert that the transference of energy in any isotropic electrical 
conductor always takes place across the lines of conduction cur- 
rent, and not merely in the case of a core uniformly lapped 
with wire, where it is nearly self-evident that it must be so. This 
is a very important result, being the post-finger pointing to a clear 
understanding of electromagnetism. Passing to the case of a 
very long straight round wire supporting an electric current, we 
are bound to conclude that the transference of energy takes place 
transversely, not longitudinally ; that is, across the wire instead 
of along it. The source of energy must, therefore, first supply 
the dielectric surrounding the wire before the substance of the 
wire itself can be influenced ; that is, the dielectric must be 
the real primary agent in the electromagnetic phenomena con- 
nected with the electric current in the wire. 

Beyond this transverse transference of energy, there does 
not, however, at first sight, appear to be much analogy between 
the case of the solenoid with a core and the straight wire in a 
dielectric. The source of energy in one case is virtually brought 
right up to the surface of the core in a uniform manner. But in 
the other case the source of energy the battery, for instance 
may be miles away at one end of the wire, and there is no 
immediately obvious uniform application of the source to the 
skin of the wire. But observe that in the former case the 
magnetic force is axial, and the electric current circular, whilst 
in the latter case the electric current (in the straight wire) is 
axial and the magnetic force circular. Now an examination 
of the electromagnetic equations shows that the conditions of 
propagation of axial magnetic force and circular current are the 


same as those for axial current and circular magnetic force. 
We therefore further conclude that (with the exchanges made) 
the phenomena concerned in the core of the solenoid and in 
the long straight wire are of the same character. 

Furthermore, if we go into detail, and consider the influence 
of the surroundings of the wire (which go to determine the 
value of the inductance of the circuit) we shall find that not 
only is the character of the phenomena the same, but that they 
may be made similar in detail (so as to be represented by 
similar curves, for example). 

The source of energy, therefore, is virtually transferred 
instantly from its real place to the whole skin of the wire, over 
which it is uniformly spread, just as in the case of the con- 
ducting core within a solenoid. 

16. So far we can go by Ohm's law and Faraday's law of 
E.M.F., and, if need be, Poisson's law of induced magnetisa- 
tion (or its modern equivalent practically). But it is quite 
impossible to stop here. Even if we had no knowledge of 
electrostatics and of the properties of condensers, we should, 
by the above course of inquiry, be irresistibly led to a theory 
of transmission of electrical disturbances through a medium 
surrounding the wire, instead of through the wire. Maxwell's 
theory of dielectric displacement furnishes what is wanted to 
explain results which are in some respects rather unintelligible 
when deduced in the above manner without reference to elec- 
trostatic phenomena. 

We learn from it that the battery or other source of energy 
acts upon the dielectric primarily, producing electric displace- 
ment and magnetic induction ; that disturbances are propa- 
gated through the dielectric at the speed of light ; that the 
manner of propagation is similar to that of displacements and 
motions in an incompressible elastic solid; that electrical 
conductors act, as regards the internal propagation, not as 
conductors but rather as obstructors, though they act as con- 
ductors in another sense, by guiding the electromagnetic 
waves along definite paths in space, instead of allowing them 
to be immediately spread away to nothing by spherical en- 
largement at the speed of light ; that when we deal with 
steady states, or only slowly varying states, involving immensely 


great wave length in the dielectric, the resulting magnetic 
phenomena are just such as would arise were the speed 
of propagation infinitely great instead of being finite ; that 
if we make our oscillations faster, we shall begin to get 
signs of propagation in the manner of waves along wires, with, 
however, great distortion and attenuation by the resistance of 
the wires ; that if we make them much faster we shall obtain 
a comparatively undistorted transmission of waves (as in long- 
distance telephony over copper wires of low resistance) ; and 
that if we make our oscillations very fast indeed, we shall have 
practically mere skin conduction of the waves along the wires 
at the speed of light (as in some of Lodge's lightning-conductor 
experiments, and more perfectly with Hertzian waves). 

Now all these things have been worked out theoretically, and, 
as is now well known, most of them have been proved experi- 
mentally ; and yet I hear someone say that Hertz's experiments 
don't prove anything in particular ! 

Lastly, from millions of vibrations per second, proceed to 
billions, and we come to light (and heat) radiation, which are, 
in Maxwell's theory, identified with electromagnetic disturb- 
ances. The great gap between Hertzian waves and waves of 
light has not yet been bridged, but I do not doubt that it will 
be done by the discovery of improved methods of generating 
and observing very short waves. 




Electric and Magnetic Force ; Displacement and Induction ; 
Elastivity and Permittivity, Inductivity and Reluctivity. 

20. Our primary knowledge of electricity, in its quantita-, 
tive aspect, is founded upon the observation of the mechanical 
forces experienced by an electrically charged body, by a mag- 
netised body, and by a body supporting electric current. In 
the study of these mechanical forces we are led to the more 
abstract ideas of electric force and magnetic force, apart from 
electrification, or magnetisation, or electric current, to work 
upon and ^produce visible effects. The conception of fields 
of force nr J irally follows, with the mapping out of space 
by means o. -ines or tubes of force definitely distributed. A 
further and very important step is the recognition that the two 
vectors, electric force and magnetic force, represent, or are 
capable of measuring, the actual physical state of tl^e medium 
concerned, from the electromagnetic point of view, when taken 
in conjunction with other quantities experimentally recognis- 
able as properties of matter, showing that different substances 
are affected to different extents by the same intensity of electric 
or magnetic force. Electric force is then to" be conceived as 
producing or being invariably associated with a flux, the electric 
displacement ; and similarly magnetic force as producing a 
second flux, the magnetic induction. 

If E be the electric force at any point and D the displace- 
ment, we have 

D = cE: (1) 


and similarly, if H be the magnetic force and B the induction, 

B = /*H (2) 

Here the ratios c and p represent physical properties of the 
medium. The one (ft), which indicates capacity for supporting 
magnetic induction, is its inductivity ; whilst the other, indi- 
cating the capacity for psrmitting electric displacement, is its 
permittivity (or permittancy). Otherwise, we may write 

E^-iD, (3) 

H = ^B; (4) 

and now the ratio c" 1 is the elastivity and fir 1 is the reluctivity 
(or reluctancy). Sometimes one way is preferable, sometimes 
the other. 

Electric and Magnetic Energy. 

21. All space must be conceived to be filled with a medium 
which can support displacement and induction. In the former 
aspect only it is a dielectric. It is, however, equally necessary 
to consider the magnetic side of the matter, and we may, with- 
out coining a new word, generally understand by a dielectric a 
medium which supports both the fluxes mentioned. 

Away from matter (in the ordinary sense) the medium con- 
cerned is the ether, and p and c are absolute constants. The 
presence of matter, to a first approximation, merely alters the 

value of these constants. The permittivity is al ys increased, 

far as is known. On the other hand, the inductivity may be 
either increased or reduced, there being a very small increase 
or decrease in most substances, but a very large increase in a 
few, the so-called magnetic metals. The range within which 
the proportionality of flux to force obtains is then a limited one, 
which, however, contains some important practical applications. 

When the fluxes vary, their rates of increase B and D are the 
velocities corresponding to the forces E and H, provided no 
other effects are produced. The activity of E is ED, and 
that of H is HB. The work spent in producing the fluxes 
(not counting what may be done simultaneously in other ways) 
is, therefore 

U = TE dD, T =| B H rfB, ... (5) 


where U is the electric and T the magnetic energy per unit 

When /* and c are constants, these give 

U - JED - JcE*, ..... (6) 
T = JHB = J/JP, ..... (7) 

to express the energy stored in the medium, electric and mag- 
netic respectively. 

When p and c are not constants, the previous expressions (5) 
will give definite values to the energy provided there be a de- 
finite relation between a force and a flux. If, however, there 
be no definite relation (which means that other circumstances 
than the value of the force control the value of the flux), the 
energy stored will not be strictly expressible in terms simply 
of the force and the flux, and there will be usually a waste of 
energy in a cyclical process, as in the case of iron, so closely 
studied by Ewing. This does nob come within the scope of a 
precise mathematical theory, which must of necessity be a sort 
of skeleton framework, with which complex details have to be 
separately adjusted in the most feasible manner that presents 

Eolotropic Relations. 

22. But a precise theory nevertheless admits of consider- 
able extension from the above with //, and c regarded as scalar 
constants. All bodies are strained -more or less, and are thereby 
usually made eolotropic, even if they be not naturally eolo- 
tropic. The force and the flux are not then usually concurrent, 
or identical in direction. But at any point in an eolotropic 
substance there are always (if force and flux be proportional) 
three mutually perpendicular axes of concurrence the prin- 
cipal axes when we have (referring to displacement) 

if the c's are the principal permittivities, the E's the corre- 
sponding effective components of the electric force, and the 
D's those of the displacement. If E be parallel to a principal 
axis, so is D. In general, by compounding the force compo- 
nents, we obtain E the actual force, and by compounding the 
flux components obtain D the displacement to correspond, 
which can only concur with E in the above-mentioned special 


cases if the principal permittivities be all different. But should 
a pair be equal, then E and D concur in the plane containing 
the equal permittivities, for the permittivity is the same for 
any axis in this plane. 

The energy stored is still half the scalar product of the \ 
force and the flux, or JED, understanding that the scalar 
product of two vectors, which is the product of their tensors 
(or magnitudes) when they concur, is the same multiplied by 
the cosine of their included angle in the general case. 

Vector-analysis is, I think, most profitably studied in the 
concrete application to physical questions, for which, indeed, it 
is specially adapted. Nevertheless, it will be convenient, a 
little later, to give a short account of the very elements of the 
subject, in order not to have to too frequently interrupt our 
electromagnetic arguments by mathematical explanations. In 
the meantime, consider the dielectric medium further. 

Distinction between Absolute and Relative Permittivity or 

23. The two quantities c and /* are to be regarded as 
known data, given over all space, usually absolute constants ; 
but when the simpler properties of the ether are complicated 
by the presence of matter, then varying in value from place 
to place in isotropic but heterogenous substances ; or, in case 
of eolotropy, the three principal values must be given, as well 
as the direction of their axes, for every point Considered. 
Keeping to the case of isotropy, the ra^'os c/c and /X//A O of the 
permittivity and inductivity of a body to that of the standard 
ether are the specific inductive capacities, electric and magnetic 
respectively ; and are mere numerics, of course. They do not 
express physical properties themselves except in the limited 
sense of telling us how many times as great something is in 
one case than in another. This is an important point. It is 
like the difference between density and specific gravity. It is 
possible to so choose the electric and magnetic units that /*= 1, 
c=l in ether; then /A and c in all bodies are mere numerics. 
But although this system (used by Hertz) has some evident 
recommendations, I do not think its adoption is desirable, at 
least at present. I do not see how it is possible for any 
medium to have less than two physical properties effective in 


the propagation of waves. If this be admitted, I think it may 
also be admitted to be desirable to explicitly admit their exist- 
ence and symbolise them (not as mere numerics, but as physical 
magnitudes in a wider sense), although their precise interpre- 
tation may long remain unknown. 

If, for example, H be imagined to be the velocity of a sub- 
stance, then JfiH 2 is its kinetic energy, and /x its density. 
And if be a torque, then c" 1 (the elastivity) is the corre- 
sponding coefficient of elasticity, the rigidity, or gwm'-rigidity, 
as the case may be ; whilst c is the coefficient of compliance, or 
the compliancy ; and JcE 2 is the stored energy of the strain. 

Dissipation of Energy. The Conduction-current ; Conduc- 
tivity and Resistivity. The Electric Current. 

24. Besides influencing the values of .the ether constants w 
above described, we have also to admit that in certain kinds 
of matter, when under the influence of electric force, energy is 
dissipated continuously, besides being stored. These are called 
electrical conductors. When the conduction is of the simplest 
(metallic) type, the waste of energy takes place at a rate pro- 
portional to the square of the electric force. Thus, if Q x be 
the Joulean waste, ->r 

Q^/jE^EC, (8) 

if C = E (9) 

This new flux C is the conduction current, and k is the conduc- 
tivity (electric). Its reciprocal is the resistivity. 

The termination -ivity is used in connection with specific 
properties. It does not always sound well at first, but that 
wears off. Sometimes the termination -ancy does as well. 

The conductivity Jc is constant (at one temperature), or is a 
linear operator, as in the previous cases with respect to //, and 
c. The dissipation of energy does not imply - its destruction, 
but simply its rejection or waste, so far as the special electro- 
magnetic affairs we are concerned with. The conductor is 
heated, and the heat is radiated or conducted away. This is 
also (most probably) an electromagnetic process, but of a 
different order. Only in so far as the effect of the heat alters 
the conductivity, &c., or, by differences of temperature, causes 


thermo-electric force, are we concerned with energy wasted 
according to Joule's law. 

The activity of the electric force, when there is waste, as well 
as storage, is 

E(C + 6) = Q 1 + U (10) 

The sum + D is the electric current, when the medium is at 
rest. When it is in motion, a further term has sometimes to 
be added, viz., the convection current. 

Fictitious Magnetic Conduction-current and Real Magnetic 


25. If a substance were found which could not support 
magnetic force without a continuous dissipation of energy, such 
& substance would (by analogy) be a magnetic conductor. Let, 
for instance, 

K = <?H, (11) 

then K is the density of the magnetic conduction current, and 
the rate of waste of energy is 

Q 2 = HK = <7H 2 (12) 

The activity of the magnetic force is now 

H(K + B) = Q 2 + T (13) 

Compare this equation with (10). The magnetic current is 
K + B. 

As there is (I believe) no evidence that the property sym- 
bolised by g has any existence, it is needless to invent a special 
name for it or its reciprocal, but to simply call g the magnetic ' 
conductivity. The idea of a magnetic current is a very useful 
one, nevertheless. The magnetic current B is of course real ; 
it is the part K that is speculative. It plays an important 
part in the theory of the transmission of waves in conductors. 

Forces and Fluxes. 

26. So far we have considered the two forces, electric and 
magnetic, producing four fluxes, two involving storage and two 
waste of energy, and we have defined the terminology when the 
state of things at a point is concerned. We reckon forces per 


unit length, fluxes per unit area, and energies or wastes per 
unit volume. It is thus a unit cube that is referred to, whose 
edge, side, and volume are utilised. But a unit cube does not 
mean a cube whose edge is 1 centim. or any other concrete 
length ; it may indeed be of any size if the quantities concerned 
are uniformly distributed throughout it, but as they usually 
vary from place to place, the unit cube of reference should be 
imagined to be infinitely small. The next step is to display the 
equivalent relations, and develop the equivalent terminology, 
when any finite volume is concerned, in those cases that admit 
of the same simple representation in the form of linear equa- 

Line-integral of a Force. Voltage and Gaussage. 

27. The line-integral of the electric force from one point to 
another along a stated path is the electromotive force along 
that path ; this was abbreviated by Fleemiug Jenkin to E.M.F. 
He was a practical man, as well as a practician. When ex- 
pressed in terms of a certain unit called the volt, electromotive 
force may be, and often is, called the voltage. This is much 
better than " the volts." I think, however, that it may often 
be conveniently termed the voltage irrespective of any par- 
ticular unit. We might put it in this way. Volta was a 
distinguished man who made important researches connected 
with electromotive force, which is, therefore, called voltage, 
whilst a certain unit of voltage is called a volt. At any rate, 
we may try it and see how it works. 

The line-integral of the magnetic force from one point to 
another along a stated path is sometimes called the magneto- 
motive force. The only recommendation of this cumbrous 
term is that it is correctly correlated with the equally cum- 
brous electromotive force. Magnetomotive force may be called 
the gaussage [pr. gowsage], after Gauss, who distinguished him- 
self in magnetic researches; and a certain unit of gaussage 
may be called a gauss [pr. gowce]. I believe this last has 
already been done, though it has not been formally sanctioned. 
Gaussage may also be experimented with. 

The voltage or the gaussage along a line is the sum of the 
effective electric or magnetic forces along the line ; the effec- 
tive force being merely the tangential component of the real 


force. Thus, electric force is the voltage per unit length, and 
magnetic force the gaussage per unit length along lines of force. 

Surface-integral of a Flux. Density and Intensity. 

28. Next as regards the fluxes, when considered with 
reference to any area. The flux through an area is the sum 
of the effective fluxes through its elementary units of area ; 
the effective flux being the normal component of the flux or 
the component perpendicular to the area. We do not, I think, 
need a number of new words to distinguish fluxes through a 
surface from fluxes per unit surface. Thus, we may speak of 
the induction through a surface (or through a circuit bounding 
it) ; or of the current through a surface (as across the section of 
a wire); or of the displacement through a surface (as in aeon- 
denser), without any indefiniteness, meaning in all cases the 
surface integral of the flux in question. 

In contradistinction to this, it may be sometimes convenient 
to speak of the density of the current, or of the induction, or 
of the displacement, that is, the amount per unit area. 
Similarly, we may sometimes speak of the intensity of the 
electric or 'magnetic force, using "density" for a flux and 
" intensity " for a force. 

It may be observed by a thoughtful reader that there is a 
good deal of the conventional in thus associating one set of 
vectors with a line, and another set with a surface, and other 
quantities with a volume. It is, however, of considerable prac- 
tical utility to carry out these distinctions, at least in a mathe- 
matical treatment. But it should never be forgotten that 
electric force, equally with displacement, is distributed through- 
out volumes, and not merely along lines or over areas. 

Conductance and Resistance. 

29. Conductivity gives rise to conductance, and resistivity 
to resistance. For explicitness, let a conducting mass of any 
shape be perfectly insulated, except at two places, A and B, to 
be conductively connected with a source of voltage. Let the 
voltage established between A and B through the conductor be 
V, and let it be the same by any path. This will be the case 
when the current is steady. Also let G be this steady current, 
in at A and out at B. We shall have 


V = RC, C = KV, ..... (1) 

where R and K are constants, the resistance and the conduct- 
ance, taking the place of resistivity and conductivity, when 
voltage takes the place of electric force, and the current that 
of current-density. The activity of the impressed voltage is 

VC = RC 2 = KV 2 ..... . . (2) 

and represents the Joulean waste per second in the whole con- 
ductor, or the volume-integral of EC or of &E 2 before con- 

Permittance and Elastance. 

30. Permittivity gives rise to permittance, and elastivity 
to elastance. To illustrate, for the conductor, substitute a 
nonconducting dielectric, leaving the terminals and external 
arrangements as before. We have now a charged condenser. 
Displacement, i.e., the time-integral of the current, takes the 
place of current in the last case, and we now have 

D = SV, V = S-!D, .... (3) 

if D is the displacement, S the permittance, and its reciprocal 
the elastance of the condenser. This elastance has been called 
the stiffness of the condenser by Lord Rayleigh. It is the 
elastic resistance to displacement. The displacement is the 
measure of the charge of the condenser. 
The total energy in the condenser is 


i.e., half the product of the force (total) and the flux (total), 
between and at the terminals ; it is also the volume-integral of 
the energy-density, or J2cE 2 . 

As the dielectric is supposed to be a non-conductor, the cur- 
rent is I) or SV, and only exists when the charge is varying. 
But it may also be conducting. If so, let the conductance be 
K, making the conduction current be C = KV. The true cur- 
rent (that is, the current) is now the sum of the conduction 
and displacement currents. Say, 


This is the characteristic equation, of a condenser. It comes 
to the same thing if the condenser be non-conducting, but be 
shunted by a conductance, K. In a conducting dielectric the 
permittivity and the conductivity are therefore in parallel arc r 
as it were. It was probably by a consideration of conduction 
in a leaky condenser that Maxwell was led to his inimitable 
theory of the dielectric, by which he boldly cut the Gordian 
knot of electromagnetic theory. 

The activity of the terminal voltage we find by multiplying, 
(5) by V, giving 


2 + SV 2 

\ .... (6) 

representing the waste in Joulean heating and the rate of in- 
crease of the electric energy. Each of these quantities is the 
sum of the same quantities per unit volume throughout the 
substance concerned. 

Permeance, Inductance and Reluctance. 

31. Permeability gives rise to permeance, inductivity to 
inductance, and reluctivity to reluctance. 

The formal relation of reluctance to reluctivity with mag- 
netic force and induction, is the same as that of resistance to 
resistivity with electric force and conduction current, or of 
elastance to elastivity with electric force and displacement. 

Permeance is the reciprocal of reluctance. In this sense I 
have used it, though only once or twice. Prof. S. P. Thomp- 
son has also used tfae word in this sense in his Cantor Lectures 
with good effect. 

If we replace our illustrative conductor by an inductor, 
supporting magnetic induction, and suppose it surrounded by 
imaginary matter of zero inductivity, and have an impressed 
gaussage instead of voltage at the terminals, we shall have a 
flux of induction which will, if the force be weak enough, vary 
as the force. If H be the gaussage and B the induction enter- 
ing at the one and leaving at the other terminal, the ratio 
H/B is the reluctance, and the reciprocal B/H is the permeance. 
The energy stored is 


When the relation of flux to force is not linear, we can still 
usefully employ the analogy with conduction current or with 
displacement by treating the ratio B/H as a function of H or 
of B ; as witness the improved and simplified way of consider- 
ing the dynamo in recent years. I must, however, wonder at 
the persistence with which the practicians have stuck to " the 
lines," as they usually term the flux in question. 

I am aware that the use of the name induction for this flux, 
which I have taken from Maxwell, is in partial conflict with an 
older use. But it is seldom, if ever, that these uses occur to- 
.gether, for one thing ; another thing is that the older (and 
often vague) use of the word induction has very largely ceased 
of late years. It was not without consideration that induction 
was adopted and, to harmonise with it, inductance and induc- 
tivity were coined. 

Inductance of a Circuit. 

32. The meaning of inductance has sometimes been mis- 
conceived. It is not a synonym for induction, nor for self-induc- 
tion, but means " the coefficient of self-induction," sometimes 
abbreviated to " the self-induction." It is essentially the same 
as permeance, the reciprocal of reluctance, but there is a prac- 
tical distinction. Consider a closed conducting circuit of one 
turn of wire, supporting a current C r As will later appear, 
this G! is also the gaussage. That is, the line -integral of the 
magnetic force in any closed circuit (or the circuitation of the 
force) embracing the current once is C r Let also B A be the 
induction through the circuit of C r Then 

BX-LA, ...... (8) 

where, by what has already been explained, LH is the perme- 
ance of the magnetic circuit, a function of the distribution of 
inductivity and of the form and position of the conducting core. 
The magnetic energy is 

JBA-iW ..... (9) 

by using the first expression in (7), remembering that H 
there is now represented by C^ and then using (8). This 


energy " of the current " resides in all parts of the field, only 
(usually) a small portion occupying the conductor itself. 

Now substitute for the one turn of wire a bundle of wires, 
N in number, of the same size and form. Disregarding small 
differences due to the want of exact correspondence between 
the bundle and the one wire, everything will be the same as 
before if the above Cj means the total current in the bundle. 
But if the same current be supported by each wire, practical 
convenience in respect to the external connections of the coil 
requires us to make the current in each wire the current. Let 
this be C, so that Cj = NO. Then we shall have, by (8), 

B i = (L 1 N)C . . . . (8a) 
to express the flux of induction ; and by (9) and (Sa), 

...... (9o) 

if L = N 2 L 1} to express the energy. This L is the inductance 
of the coil. It is N 2 times the permeance of the magnetic 

Again, regarding the coil as a single circuit, B : N is the 
induction through it that is, B x through each winding. Calling 
this total B, we have, by (8a), 

B = B 1 N=(L 1 N 2 )C = LC, .... (86) 

which harmonises properly with (9a). 

The difference between inductance and permeance, therefore, 
merely depends upon the different way of reckoning the current 
in the coil. With one winding only, they are identical. I 
should here observe that I am employing at present rational 
units. Their connection with the Gaussian units will appear 
later. It would only serve to obscure the subject to bring in 
47T, that arbitrary and unnecessary constant which has puzzled 
so many people. 

It will be seen that the distinction between permeance and 
inductance is a practical necessity, in spite of their fundamental 
identity. But which should be which ? On the whole, I 
prefer it as above stated, especially to connect with self-induc- 
tion. Regarding permeability itself, it would seem that this 
name is more particularly suitable to express the ratio /*//z of 


the inductivity of a medium to that of ether, which is, in fact r 
consistent with the original meaning, I believe, as used by 
Sir W. Thomson in connection with his " electromagnetic defi- 
nition " of magnetic force. But to inductivity, as before- 
mentioned, a wider significance should be attached. As has 
been more particularly accentuated by Prof. Riicker, we really 
do not know anything about the real dimensions of ft and c ; 
or, more strictly, we do not know the real nature of the 
electromagnetic mechanism, so that ft and c are very much 
what we choose to make them, by assumptions. The two prin- 
cipal systems are the so-called electrostatic, in which c = 1 in 
ether, and the electromagnetic, in which ft = 1 in ether. But 
with these specialities we have no further concern at present. 

Cross-connections of Electric and Magnetic Force. Circuital 
Flux. Circuitation. 

33. The two sets of quantities, the electric and magnetic 
forces, with their corresponding fluxes and currents, and the 
connected products and ratios, may be considered quite inde- 
pendently of one another, without any explicit connection being 
stated between the electric set and the magnetic set, whether 
they coexist or not. But to have a dynamical electromagnetic 
theory, we require to know something more, viz., the cross- 
connections or interactions between E and H. Or, in another 
form, we require to know how an electric field and a magnetic- 
field mutually influence one another. 

One of these interactions has been already partially men- 
tioned, though only incidentally, in stating the meanings of 
permeance and inductance. It was observed that the electric 
current in a simple conductive circuit was measured by the 
gaussage in the corresponding magnetic circuit. 

A word has been much wanted to express in a convenient 
and concise manner the property possessed by some fluxes and 
other vectors of being distributed in closed circuits. This want 
has been recently supplied by Sir W. Thomson's introduction of 
the word " circuital " for the purpose.* Thus electric current is 
a circuital flux, and so is magnetic induction. The fundamental 
basis of the property is that as much of a circuital flux enters 

* "Mathematical and Physical Papers," Vol. III., p. 451. 


any volume at some parts of its surface as leaves it at others, 
so that the flux has no divergence anywhere. This qualification, 
"anywhere," should be remembered, for a flux which may 
diverge locally, as, for instance, electric displacement, is not 
circuital in general, though even electric displacement may be 
circuital sometimes. Further, as a flux need not be distributed 
throughout a volume, but may be confined to a surface, or to 
a line, we have then specialised meanings of circuital and of 
divergence. Or a volume-distribution and a surface or line- 
distribution of a flux may be necessarily conjoined, without, 
however, any departure from the essential principle concerned. 
The word " circuital," which will be often used, suggested to 
me the word " circulation," to indicate the often-occurring 
operation of a line-integral in a closed circuit ; as, for instance, 
in the estimation of circuital voltage or gaussage. Now, in the 
case of a moving fluid, Sir W. Thomson called the line-integral 
of the velocity in a closed circuit the " circulation." This is 
curiously like " circulation." But " circulation " seems to have 
too specialised a meaning to be suitable for application to any 
vector, and I shall employ " circulation." The operation of 
circuitation is applicable to any vector, whether it be circuital 
or not. 

First Law of Circuitation. 

34. Now in the case of a simple conductive circuit, we have 
two circuital fluxes. There is a circuital conducting core sup- 
porting an electric current, and there is a circuital flux of in- 
duction through the conductive circuit. In the electric circuit 
we have Ohm's law, 

E = RC, (1) 

where E is the circuital voltage, C the current, and R the re- 
sistance. And in the magnetic circuit we have a formally 
similar relation, 

H=L-IB, (2) 

where H is the circuital gaussage, B the induction, and Lr 1 the 
reluctance. Or, 

B = LH, (3) 

where L is the inductance (or the permeance, when there is 
only one turn of wire). 



Now, the cross-connection in this special case is implied in 
the assertion that H and G are the same quantity, when mea- 
sured in rational units The expression of the law of which 
this is an illustration is contained in any of the following alter- 
native statements. 

The line-integral of the magnetic force in any closed circuit 
measures the electric current through any surface bounded by 
the circuit. Or, 

The circuitation of the magnetic force measures the electric 
current through the circuit. Or, 

The electric current is measured by the magnetic circuita- 
tion, or by the circuital gaussage. 

The terminology of electromagnetism is in a transitional 
state at present, owing to the change that is taking place in 
popular ideas concerning electricity, and the unsuitability of the 
old terminology, founded upon the fluidity of electricity, for a 
comprehensive view of electromagnetism. This is the excuse 
for so many new words and forms of expression. Some of 
them may find permanent acceptation. 

The above law applies to any circuit of any size or shape, 
and irrespective of the kind of matter it passes through, mean- 
ing by " circuit " merely a closed line, along which the gauss- 
age is reckoned. By " the current " is to be understood the 
current ; not merely the conduction current alone, or the dis- 
placement current alone, but their sum (the convection current 
term will be considered separately). 

It is also necessary to understand that a certain convention 
is implied in the statement of the law, regarding positive 
senses of translation and rotation when taking line and surface 
integrals. Look at the face of a watch, and imagine its circum- 
ference to be the electric circuit. The ends of the pointers 
travel in this circuit in the positive sense, if you are looking 
through the circuit along its axis in the positive sense. Also, you 
are looking at the negative side of the circuit. Thus, when the 
current is positive in its circuit, the magnetic induction goes 
through it in the positive direction, from the negative side to the 
positive side. Otherwise, the positive sense of the current in a 
circuit and the induction through it are connected in the same 
way as the motions of rotation and translation of a nut on an 
ordinary right-handed screw. This is the " vine " system used 


by all British writers; but some continental writers use the 
"hop" system, in which the rotation is the other way, for the 
same translation. It is useless trying to work both systems, 
and when one comes across the. left-handed system in papers, 
it is, perhaps, best to marginally put the matter straight, and 
then ignore the text. 

Second Law of Circuitation. 

35. The other cross-connection required is a precisely 
similar relation between voltage and magnetic current, with, 
however, a change of sign. Thus : 

The negative line-integral of the electric force in any circuit 
(or the electric circuitation) measures the magnetic current 
through the circuit. Or, 

The voltage in any circuit measures the magnetic current 
through the circuit taken negatively. Or, 

Magnetic current is measured by the circuital voltage re- 
versed ; and other alternative equivalent statements. 

Definition of Curl. 

36. In the above laws of circuitation the currents are the 
concrete currents (surface-integrals), and the forces also the 
concrete voltage or gaussage. When we pass to the unit 
volume it is the current-density that is the flux. The circuita- 
tion of the force is then called its " curl." Thus, if J be the 
electric current and G the magnetic current, the two laws are 

curlH^J, (4) 

-cur!E 1 = G, (5) 

where E x and H x are the electric and magnetic force of the 
field. We may now say concisely that 

The electric current is the curl of the magnetic force. 

The magnetic current is the negative curl of the electric force. 

There is nothing transcendental about " curl." Any man 
who understands the laws of circuitation also understands 
what " curl " means, though he may not himself be aware of 
his knowledge, being like the Frenchman who talked prose for 
many years without knowing it. The concrete circuitation is 
sufficient for many problems, especially those concerning linear 



conductors in magnetic theory. But it does not suffice for 
mathematical analysis, and to go into detail we require to pass 
from the concrete to the specific and use curl. How to mani- 
pulate " curl " is a different matter altogether from clearly 
understanding what it means and the part it plays. The latter 
is open to everybody ; for the former, vector-analysis is most 

Let a unit area be chosen perpendicular to the electric cur- 
rent J. Its edge is then the circuit to which H a belongs in (4). 
The gaussage in this circuit measures the current-density. 
Similarly, regarding (5), the voltage in a unit circuit perpen- 
dicular to the magnetic current measures its density (nega- 
tively). In short, what circuitation is in general, curl is the 
same per unit area. 

Impressed Force and Activity. 

37. In the statement of the laws of circuitation, I have 
intentionally omitted all reference to impressed forces. That 
there must be impressed forces is obvious enough, because a 
dynamical system comprehending only the electric and mag- 
netic stored energies and the Joulean waste, is only a part of 
the dynamical system of Nature. We require means of show- 
ing the communication of energy to or from our electromagnetic 
system without having to enlarge it by making it a portion of 
a more complex system. Thus, taking it as it stands at pre- 
sent, the activity per unit volume we have seen to be 

..... (6) 

where the left side expresses the activity of the electric and the 
magnetic force on the corresponding currents, and the right side 
what results, viz., waste of energy, Q per second, and increase per 
second of the electric energy U and the magnetic T ; and, as 
there are supposed to be no impressed forces, if we integrate 
through all space, we shall obtain 


where 2 means summation of what follows it. Or, if Q be the 
total waste, and similarly U and T the total energies, . 


meaning, that whatever energy there be wasting itself is derived 
solely from the electric or magnetic energy, which decrease 
accordingly. This is the persistence of energy when there are 
no impressed forces. 

Now, if there be impressed forces communicating energy at 
the rate A, the last equation must become 

T , .... (8) 

and A must be the sum of the activities of the impressed forces 
f in the elements of volume, in whatever way space may be 
divided into elements, large or small, and however we may 
choose to reckon the impressed forces. There may be many 
ways of doing it ; f may sometimes, for example, be an ordi- 
nary force, and v, the velocity to match, is then a translational 
velocity. But for our immediate purpose, it is naturally con- 
venient to reckon the impressed forces electrically and mag- 
netically ; so that the corresponding velocities are the electric 
and magnetic currents. We shall then have, if e be the im- 
pressed electric, and h the impressed magnetic force, 

to represent their activity per unit volume, and in all space, 

... (9) 

Instead of (7). This is the integral equation of activity. We 
cannot remove the sign of summation and make the same form 
do for the unit volume, for this would make every unit volume 
independent of the rest, and do away with all mutual action 
between contiguous elements and transfer of energy between 
them. This matter will be returned to in connection with the 
transference of energy. 

Distinction between Force of the Field and Force of the 


38. The distinction between H x and H and between E x and 
E is often a matter of considerable importance. We have 



Now it is E and H that are effective in producing fluxes. Thus 
E is the force of the flux D and also of ; and H is the force 
of the flux B. On the other hand, in the laws of circuitation, 
as above expressed, the impressed forces do not count at all j 
so that we have, in terms of the forces E and H, 

curl(H-h) = J, (12) 

-curl (E-e) = G, (13) 

equivalent to (4) and (5). To distinguish from the forces of 
the fluxes, I sometimes call E l and H 3 the forces " of the field." 
Of course they only differ where there is impressed force. As 
the distribution of the energy, as well as of the fluxes, depends 
upon E and H, it is usually best to use them in the formulae. 

Classification of Impressed Forces. 

39. The vectors representing impressed electric and mag- 
netic force demand consideration as to the different forms they 
may assume. Their line-integrals are impressed voltage and 
gaussage. Their activities or powers are eJ and hG- respectively 
per unit volume, and in this statement we have a sort of defi- 
nition of what is to be understood by impressed force. For, 
J being the electric current anywhere, if there be an impressed 
force e acting, the amount eJ of energy per unit volume is 
communicated to, or taken in by, the electromagnetic system 
per second ; and this should be understood to take place at the 
spot in question. It must then be either stored on the spot, 
or wasted on the spot, or be somehow transmitted away to other 
places, to be there stored or wasted, according to a law which 
will appear later on. Similarly as regards h and G-. 

But this concerns only the reckoning of impressed force, and 
is independent of its physical origin, which may be of several 
kinds. Thus under e we include 

(1.) Voltaic force. 
(2.) Thermo-electric force. 
(3.) The force of intrinsic electrisation. 
(4.) Motional electric force. 

(5.) Perhaps due to various secondary causes, especially ia 
connection with strains. 


And under h we include 

(1.) The force of intrinsic magnetisation. 

(2.) Motional magnetic force. 

(3.) Perhaps due to secondary causes. 

Voltaic Force. 

40. Voltaic force has its origin in chemical affinity. This 
is still a very obscure matter. For a rational theory of Chemistry, 
one of the oldest of the sciences, we may have to wait long, in 
spite of the activity of chemical research and of the develop- 
ment of the suggestive periodic law. Yet Chemistry and 
Electricity are so intimately connected that we cannot under- 
stand either without some explanation of the other. Elec- 
tricity is, in its essentials, a far simpler matter than Chemistry, 
and it is possible that great light may be cast upon chemical 
problems (and molecular physics generally) by previous dis- 
coveries and speculations in Electricity. The very abstract nature 
of Electricity is, in some respects, in its favour. For there is 
considerable truth in the remark (which, if it has not been made 
before, is now originated) that the more abstract a theory is, 
the more likely it is to be true. For example, it may be that 
Maxwell's theory of displacement and induction in the ether 
is far more than a working theory, and is something very near 
the truth, though we know not what displacement and induc- 
tion are. But if we try to materialise the theory by inventing a 
special mechanism we are almost certain to go wrong, however 
useful the materialisation may be for certain purposes. No 
one knows what matter is, any more than ether. But we do 
know that the properties of matter are remarkably complex. 
It is, therefore, a real advantage to get away from matter 
when possible, and think of something far more simple and 
uniform in its properties. We should rather explain matter 
in terms of ether, than go the other way to work. 

However this be, we have the fact that definite chemical 
changes involve definite voltages, and herein lies one of the 
most important sources of electric current. Furthermore, 
there is the remarkable connection between the quantity of 
matter and the time-integral of the current (or quantity of 
electricity) produced, involved in the law of electro-chemical 


equivalents, which is one of the most suggestive facts in 
physics, and must be a necessary part of the theory of the 
atom which is to come. That the energy of chemical affinity 
may itself be partly electromagnetic is likely enough. That 
even conduction may be an electrolytic process is possible, in 
spite of the sweet simplicity of Ohm's law and that of Joule. 
For these laws are most probably merely laws of averages. 
The well-known failure of Ohm's law (apparent at any rate) 
when the periodicity of electromagnetic waves in a conductor 
amounts to billions per second may perhaps arise from the 
period being too short to allow of the averages concerned in 
Ohm's law to be established. If so, this may give a clue to the 
required modification. 

Thermo-electric Force. 

41. Thermo-electric force has its origin in the heat of bodies, 
manifesting itself at the contact of different substances or 
between parts of the same substance differing in temperature. 
Now heat is generally supposed to consist in the energy of 
agitation of the molecules of bodies, and this is constantly 
being transferred to the ether in the form of radiant energy, 
i.e., electromagnetic vibrations of very great frequency, but in 
a thoroughly irregular manner. It is this irregularity that is 
a general characteristic of radiation. Now the result of sub- 
jecting conductors to electric force is to dissipate energy and 
to heat them. This is, however, an irreversible process. But 
when contiguous parts of a body are at different temperatures, 
a differential action on the ether results, whereby a continued 
effect of a regular type is produced, reversible with the current, 
and therefore formularisable as due to an intrinsic electric 
force, the thermo-electric force. At the junction of different 
materials at the same temperature it is still the heat that is 
the source of energy. 

The theory of thermo-electric force due to Sir W. Thomson, 
based upon the application of the Second Law of Thermo- 
dynamics (the First is a matter of course) to the reversible heat 
effects has been verified for conductive metallic circuits by the 
experiments of its author, and those of Prof. Tait and others. 
With some success the same principle has also been applied by 
von Helmholtz to voltaic cells, which are thermo-electric as well 


as voltaic cells. There are wheels within wheels, and Ohm's 
law is merely the crust of the pie. 

Intrinsic Electrisation. 

42. Intrinsic electrisation is a phenomenon shown by most 
solid dielectrics under the continued action of electric force. It 
is the manifestation of a departure from perfect electric elasti- 
city, and is probably due to a molecular rearrangement, result- 
ing in a partial fixation of the electric displacement, whereby it 
is rendered independent of the " external " electrising force. 
Thus the displacement initially produced by a given voltage 
slowly increases, and upon the removal of the impressed voltage 
only the initial displacement will subside, if permitted, imme- 
diately. The remainder has become intrinsic, for the time, and 
may be considered due to an intrinsic electric force e. If I x be 
the intensity of intrinsic electrisation, and c the permittivity, 

ii- ......... a) 

Ij is the full displacement the force e can produce elastically, 
all external reaction being removed by short circuiting. It is 
not necessarily the actual displacement. The phenomenon of 
41 residual charge," "soakage," "absorption," &c., are accounted 
for by this e and its slow variations. 

Maxwell attempted to give a physical explanation of this 
phenomenon by supposing the dielectric to be heterogeneously 
conductive. This is perhaps not the most lucidly successful of 
Maxwell's speculations. How far electrolysis is concerned in 
the matter is not thoroughly clear. 

Intrinsic Magnetisation. 

43. Intrinsic magnetisation is, in some respects, a similar 
phenomenon, due to a passage from the elastic to the intrinsic 
form of induction externally induced in solid materials. Calling 
the intensity of intrinsic magnetisation I 2 , we have 

where h is the equivalent intrinsic magnetic force, and /* the 
inductivity (elastically reckoned). 

In one important respect intrinsic induction is a less general 
phenomenon than intrinsic displacement. There is no magnetic 


conductivity to produce similar results as regards the magnetic 
current as there is electric conductivity as regards the electric 
current. But if there were, then we could have a magnetic 
" condenser," with a magnetically conductive external circuit, 
and get our residual results to show themselves in it, quite 
similarly in kind to, though varying in magnitude and perma- 
nence from, what we find with an electric condenser. 

The analogue of Maxwell's explanation of " absorption " 
would be heterogeneous magnetic conductivity. This is infi- 
nitely more speculative than the other, which is sufficiently 

Swing's recent improvement of Weber's theory of magnetism 
seems important. But as in static explanations of dynamical 
phenomena the very vigorous molecular agitations are ignored, 
it is clear that we have not got to the root of the matter. We 
want another Newton, the Newton of molecular physics. Facts 
there are in plenty to work upon, and perhaps another heaven- 
born genius may come to make their meaning plain. Pro- 
perties of matter are all very well, but what is matter, and why 
their properties ? This is not a metaphysical inquiry, but con- 
cerns the construction of a physical theory. 

The Motional Electric and Magnetic Forces. Definition of a 

44. The motional electric and magnetic forces are the 
forces induced by the motion of the medium supporting the 
fluxes. To express them symbolically, it will save much and 
repeated circumlocution if we first define the vector-product 
of a pair of vectors. 

Let a and b be any vectors, and c their vector-product. This 
is denoted by 

c = Vab, (3) 

the prefix V meaning "vector," or, more particularly here, 
"vector-product." The vector c is perpendicular to the plane 
of the vectors a and b, and its tensor (or magnitude) equals 
the product of the tensor of a into the tensor of b into the 
sine of the angle between a and b. Thus 

c = absmO, (4) 

if the italic letters denote the tensors, and 6 be the included 


angle of the vectors. As regards the positive sense of the 
vector c, this is reckoned in the same way as before explained 
with regard to circuitation. Thus, when the tensor c is posi- 
tive, a positive rotation about c in the plane of a and b will 
carry a to b. If the time by a watch is three o'clock, and the 
big hand be a and the little hand b, then the vector c is 
directed through the watch from its face to its back. These 
vector-products are of such frequent occurrence, and their 
Cartesian representation is so complex, that the above concise 
way of representing them should be clearly understood. 

On this understanding, then, we can conveniently say that 
the motional electric force is the vector-product of the velocity 
and the induction, and that the motional magnetic force is the 
vector -product of the displacement and the velocity. Or, in 
symbols, according to (3), 

....... (5) 

h = VDq, ..... . . (6) 

where q. is the vector velocity. 

Example. A Stationary Electromagnetic Sheet. 

45. It should be remembered that we regard the dis- 
placement and the induction as actual states of the medium, 
and therefore if the medium be moving, it carries its states 
with it. Besides this, it usually happens that these states are 
themselves being transferred through the medium (independently 
of its translational motion), so that the resultant effect on pro- 
pagation, considered with respect to fixed space, is a combination 
of the natural propagation through a medium at rest, and what 
we may call the convective propagation. Of course we could 
not expect the two laws of circuitation for a medium at rest to 
remain true when there is convective propagation. 

The matter is placed in a very clear light by considering the 
very simple case of an infinite plane lamina of E and H travel- 
ling at the speed of light v perpendicularly to itself through a 
homogeneous dielectric. This is possible, as will appear later, 
when E and H are perpendicular, and their tensors are thus 
related : 

. (7) 


Or, vectorising v to v, 

...... (8) 

according to the definition of a vector-product, gives the 
directional relations as well as the numerical. 

Now, suppose we set the whole medium moving the other 
way at the speed of light. The travelling plane electro- 
magnetic sheet will be brought to rest in space, whilst the 
medium pours past it. Being at rest and steady, the electric 
displacement and magnetic induction can cnly be kept up by 
coincident impressed forces, viz. : 

e = E, h = H 

Now compare (8) with (5) and (6) ; consider the directions 
carefully, and remember that the velocity q. is the negative of 
the velocity v, and we shall obtain the formulae (5), (6), which 
are thus proved for the case of plane wave motion, by starting 
with a simple solution belonging to a medium at rest. 

The method is, however, principally useful in showing the 
necessity of, and the inner meaning of the motional electric 
and magnetic forces. To show the general application of (5) 
and (6) requires a more general consideration of the motional 
question, to which we now proceed. 

Connection between Motional Electric Force and 
" Electromagnetic Force." 

46. A second way of arriving at the motional electric 
force is by a consideration of the work done in moving a con- 
ducting circuit in a magnetic field. It results from Ampere's 
researches, and may be independently proved in a variety 
of ways, that the forcive (or system of forces) acting upon 
a conducting circuit supporting a current, may be accounted 
for by supposing that every element of the conductor is subject 
to what Maxwell termed " the electromagnetic force." This is 
a force perpendicular to the vector current and to the vector 
induction, and its magnitude equals the product of their ten- 
sors multiplied by the sine of the angle between them. In 
short, the electromagnetic force is the vector-product of the 
current and the induction. Or, by the definition of a vector- 

F = VCB, ...... (9) 


if P is the force per unit volume, the current, and B the 
induction. Here F is the force arising from the stress in the 
magnetic field. Its negative, say f, is therefore the impressed 
mechanical force, or 

f=VBO, ...... (10) 

to be used when we desire to consider work done upon the 
electromagnetic system. 

The activity of f is fq, if q be the velocity ; or, by (10), 

fq = qVBC (11) 

This is identically the same as 

fq = CVqB, ..... (12) 

by a fundamental formula in vector-analysis.* Here, on the 
left side, the activity is expressed mechanically ; on the right 
side, on the other hand, it is expressed electrically, as the 
scalar product of the current and another vector, which is the 
corresponding force; it is necessarily an electric force, and 
necessarily impressed. So, calling it e, we have 

e = VqB (13) 

again, to express the motional electric force. 

It should be observed that we are not concerned in this 
mode of reasoning with the explicit connection between e and C; 
and in this respect the process is remarkably simple. As it, 
however, rests upon a knowledge of the electromagnetic force, 
we depart from the method of deriving relations previously 
pursued. But, conversely, we may by (11) and (12) derive the 
electromagnetic force from the motional electric force. 

Variation of the Induction through a Moving Circuit. 

47. A third method of arriving at (13) is by considering 
the rate of change of the amount of induction through a 
moving circuit. We need not think of a conducting circuit, 
but, more generally, of any circuit. Let it be moving in any 
way whatever, changing in shape and size arbitrarily. The 
induction through it is altering in two entirely distinct ways. 
First, there is the magnetic current before considered, due to 
the time-variation of the induction, so that, if the circuit were 

* Proved, with other working formulae, in the chapter on the Algebra 
of Vectors. 


at rest in its momentary position, we should have the second 
law of circulation true in its primitive form 

-curlE = G, (14) 

when expressed for a unit circuit. But now, in addition, the 
motion of the elements of the circuit in the magnetic field 
causes, independently of the time-variation of the field, addi- 
tional induction to pass through the circuit. Let its rate of 
increase due to this cause be g per unit area. If, then, we 
assume that the circulation of the electric force E (of the flux) 
equals the rate of decrease of the induction through the circuit 
always, whether it be at rest or in motion, the equation (14) 
becomes altered to 

-curlE = G + g, ..... (15) 

where the additional g may be regarded as a fictitious magnetic 
current. That it is also expressible as the curl of a vector is 
obvious, because it depends upon the velocity of each part of 
the circuit, and is therefore a line-integral. Examination in 
detail shows that 

g=-curlVciB, (16) 

BO that we have, by inserting (16) in (15), 

-curl(E-e) = G, .... (17) 

the standard form of the second law of circuitation, when we 
use (13) to express the impressed force. 

The method by which Maxwell deduced (13) is substantially 
the same in principle ; he, however, makes use of an auxiliary 
function, the vector-potential of the electric current, and this 
rather complicates the matter, especially as regards the physical 
meaning of the process. It is always desirable when possible 
to keep as near as one can to first principles. The above may, 
without any formal change, be applied to the case of assumed 
magnetic conductivity, when G involves dissipation of energy; 
the auxiliary g in (15), depending merely upon 'the motion of the 
circuit across the induction, does not itself involve dissipation. 

Modification. Circuit Fixed. Induction moving 


48. Perhaps the matter may be put in a somewhat clearer 
light by converting the case of a moving circuit into that of a 



oircuit at rest, and then employing the law of circuitation in 
its primitive form. The moving circuit has at any instant a 
definite position. Imagine it to be momentarily fixed in that 
position, by stopping the motion of its parts. In order that 
the relation of the circuit to the induction should be the same 
&s when it was moving, we must now communicate momentarily 
to the lines of induction the identically opposite motion to the 
(abolished) motion of the part of the circuit they touch. 

We now get equation (15), on the understanding that g means 
the additional magnetic current through a fixed circuit due to 
a given motion of the lines of induction across its boundary, 
such motion being the negative of the (abolished) motion of 
the circuit. The matter, is, therefore, simplified in treatment. 
For, in the former way, the process of demonstrating (15) which 
I have referred to as an " examination in detail," is really 
considerably complex, involving the translation, rotation, and 
distortion of an elementary circuit (or equivalently for any 
circuit). Fixing the circuit does away with this, and we have 
merely to examine what happens at a single element of the cir- 
cuit, as induction sweeps across it, in increasing the induction 
through the circuit, and then apply the resulting formula to 
every element. 

In the consideration of a single element, it is immaterial what 
the shape of the circuit may be; it may, therefore, be chosen to be 

a unit square in the plane of the paper, one of whose sides, AB, 
is the element of unit length. Now, suppose the induction at AB 
is perpendicular to the plane of the paper, directed downwards, 
and that it moves from right to left perpendicularly across 
AB. Let also from A to B be the positive sense in the circuit. 
It is evident, without any argumentation, that the directions 
chosen for q and B are the most favourable ones possible for 


increasing the induction through the circuit, and that the rate 
of its increase, so far as AB alone is concerned, is simply qB, 
the product of the tensors of the velocity q of transverse motion 
and of the induction B. Further, if the velocity q be not wholly 
transverse to B as described, but still be wholly transverse to 
AB, we must take, instead of q, the effective transverse com- 
ponent q sin 6 y if be the angle between q and B, making our 
result to be qB sin 0. Now, this is the tensor of VqB, whose 
direction is from A to B. The motional electric force in the 
element AB is therefore from B to A, and is VBq, because it 
is the negative circuitation which measures the magnetic cur- 
rent through a circuit. Lastly, if the motion of B be not 
wholly transverse to AB, we must further multiply by the 
cosine of the angle between VBq and the element AB. 
This merely amounts to taking the effective part of VBq along 
the circuit. So, finally, we see that VBq fully represents the 
impressed electric force per unit length in AB when it is fixed, 
and the induction moves across it, or that its negative 

e = VqB 

represents the motional electric force when it is the element 
AB that moves with velocity q through the induction B. Now, 
apply the process of circuitation, and we see that e is such that 
its curl represents the rate of increase of induction through the 
unit circuit due to the motion alone. 

This may seem rather laboured, but is perhaps quite as much 
to the point as a complete analytical demonstration, where one 
may get lost in the maze of differential coefficients, and have 
some difficulty in interpreting the analytical steps electromag- 

The fictitious motion of the induction above assumed has 
nothing to do with the real motion of the induction through 
the medium. If there be any, its effect is fully included in the 
term G, the real magnetic current. 

The Motional Magnetic Force. 

49. The motional magnetic force h may be similarly 
deduced. First we have the primitive form of the first law of 

curlH=J, (18) 


when the unit circuit is at rest, where J is the complete electric 
current-density, and next 

curlH = J+j, ..... (19) 

when the circuit moves ; where the auxiliary j is a fictitious 
electric current equivalent to the increase of displacement 
through the circuit by its motion only. Next show that 

j = curlh, ...... (20) 

and h = VDq, ...... (21) 

by similar reasoning to that concerning e ; so that by insertion 
in (19) the first law of circuitation is reduced to the standard 

curl (H-h) = J, ..... (22) 

with the special form of the impressed force h stated. 

Comparing the form of h with that of e we observe that 
there is a reversal of direction in the vector-products, the flux 
being before the velocity in one and after it in the other. This 
arises from the opposite senses of circuitation of the electric 
and the magnetic force to represent the magnetic and electric 

The "Magneto-electric Force." 

50. The activity of the motional h is found by multiplying 
it by the magnetic current, and is, therefore, 

by the same transformation as from (11) to (12). 

We conclude that VGD is an impressed mechanical force, 
per unit volume, and, therefore, that VDG- is a mechanical 
force, that is, of the Newtonian type, arising from the electric 
stress. By analogy with the electromagnetic force it may be 
termed the magnetoelectric force, acting on dielectrics support- 
ing displacement when the induction varies with the time. Of 
this more hereafter. 

Electrification and its Magnetic Analogue. Definition of 


51. So far nothing has been laid down about electrification. 
But the laws of circuitation cannot be completed without 
including electrification and its suggested magnetic analogue. 


Describe a closed surface in a dielectric, and observe the net 
amount of displacement leaving it. This, of course, means the 
excess of the quantity leaving over that entering it. If the 
net amount be zero, there is no electrification within the region 
bounded by the surface. If the amount be finite, there is just 
that amount of electrification in the region. This is indepen- 
dent altogether of its distribution within the region, and of 
the size and shape of the region. 

More formally, the surface-integral of the displacement 
leaving any closed surface measures the electrification within it. 

This being general, if we wish to find the distribution of 
electrification we must break up the region into smaller regions, 
and in the same manner determine the electrifications in them. 
Carrying this on down to the infinitely small unit volume, we, 
by the same process of surface-integration, find the volume- 
density of the electrification. It is then called the divergence 
of the displacement. 

That is, in general, the divergence of any flux is the amount 
of the flux leaving the unit volume. 

And in particular, the divergence of the displacement 
measures the density'^pf electrification. 

Similarly, the divergence of the induction measures the 
" magnetification," if thu;e is any to measure, which is a very 
doubtful matter indeed. There is no evidence that the flux 
induction has any divergence ; it is purely a circuital flux, so 
far as is certainly known, and this is most intimately connected 
with the other missing link in a symmetrical electromagnetic 
scheme, the (unknown) magnetic conductivity. 

Divergence is represented by div, thus : 

divD = />, (1) 

divB = o-, (2) 

if p and cr are the electrification and magnetification densities 

In another form, electrification is the source of displacement, 
and magnetification the source of induction. How these fluxes 
are distributed after leaving their sources is a perfectly indif- 
ferent matter, so far as concerns the measure of the strength of 
the sources. In an isotropic uniform medium at rest, the fluxes 
naturally spread out uniformly and radially from point-sources 


of displacement or of induction. The density of the fluxes 
then varies as the inverse square of the distance, because the 
concentric spherical surfaces through which they pass vary in 
area directly as the square of the distance. Thus 

D- ? B- * .... (3) 


are the tensors of the displacement and induction at distance r 
from point-sources p and a-. 

If the source be spread uniformly over a plane in a uniform 
isotropic medium to surface-density p or cr, then, by the mere 
symmetry, we see that half the flux goes one way and half the 
other, perpendicularly to the plane, so that 


at any distance from the plane. But if we by any means make 
the source send all the flux one way only, then 

D = />, B = o-, ..... (5) 

at any distance. 

A Moving Source equivalent to a Convection Current, and 
makes the True Current Circuital. 

52. The above being merely to concisely explain the 
essential meaning of electrification in relation to displace- 
ment, and how it is to be measured, consider a point- 
source or charge to be in motion through a dielectric at 
rest. Starting with the charge at rest at one place, the 
displacement is radial and stationary. When permanently at 
rest in another place, the displacement is the same with 
reference to it. In the transition, therefore, the displace- 
ment has changed its distribution. There must, therefore, 
be electric current. Now, the only place where the dis- 
placement diverges, however the source may be moving, is 
at the source itself, and therefore the only place where the 
displacement current diverges is at the source, because it is 
the time-variation of the displacement. The displacement 
current is therefore circuital, with the exception of a missing 



link at the moving charge. If we suppose that the charge p 
moving with velocity u constitutes a current Tip, that is, in 
the same sense as the motion, and such that the volume- 
integral of the current density is u/>, then the complete 
system of this " convection " current, and the displacement 
current together form a circuital flux. 

Thus, suppose the charge to be first outside a closed surface 
and then move across it to its inside. When outside, if the 
displacement goes through the surface to the inner region, 
it leaves it again. On the other hand, when the charge is 
inside, the whole displacement passes outward. Therefore, 
when the charge is in the very act of crossing the surface, the 
displacement through it outward changes from to /o, and this 
is the time-integral of the displacement current outward whilst 
the charge crosses. This is perfectly and simultaneously com- 
pensated by the convection current, making the whole current 
always circuital. 

The electric current is, therefore, made up of three parts, 
the conduction current, the displacement current, and the con- 
vection current ; thus, 


p being the volume-density of electrification moving through 
the stationary medium with the velocity u. 

If the medium be also moving at velocity q. referred to- 
fixed space, we must understand by u above the velocity also 
referred to fixed space. The velocities q. and u are only the 
same when the medium and the charge move together. Thus 
it will come to the same thing if we stop the motion of the 
charge altogether, and let the medium have the motion equiva- 
lent to the former relative motion. 

Similarly, if there should be such a thing as diverging in- 
duction, or the " magnetification " denoted by <r above, then we- 
shall be obliged to consider a moving magnetic charge as con- 
tributing to the magnetic current, making the complete mag- 
netic current be expressed by 

if w be the velocity of the magnetification of density o-. 


Examples to illustrate Motional Forces in a Moving Medium 
with a Moving Source. (1.) Source and Medium with a 
Common Motion. Flux travels with them undisturbed. 

53. In order to clearly understand the sense in which 
motion of a charge through a medium, or motion of the 
medium itself, or of both together with respect to fixed space, 
is to be understood, and of the part played therein by the 
motional electric and magnetic forces, it will be desirable to 
give a few illustrative examples of such a nature that their 
meaning can be readily followed from a description, without 
the mathematical representation of the results. It does not, 
indeed, often happen that this can be done with profit and 
without much circumlocution. In the present case, however, 
it is rather easier to see the meaning of the solutions from a 
description, than from the formulae. 

In the first place, let us start with a single charge p at rest 
at any point A in an infinite isotropic non-conducting dielectric 
ether, for example which is also at rest. Under these 
circumstances the stationary condition is one of isotropic radial 
displacement from the charge at A according to the inverse- 
square law, and there is nothing to disturb this distribution. 

Now, if the whole medium and the charge itself are supposed 
to have a common motion (referred to an assumed fixed space, 
in the background, as it were), no change whatever will take 
place in the distribution of displacement referred to the moving 
charge. That this should be so in a rational system we may 
conclude from the relativity of motion (the absolute motion of 
the universe being quite unknown, if not inconceivable) com- 
bined with our initial assumption that the electric flux (and 
the magnetic flux not here present) represent states of 
the medium, which may be carried with it just as states 
of matter are carried with matter in its motion. But as 
the charge, and with it the displacement, move through 
space as a rigid body without rotation, the changing dis- 
placement at any point constitutes an electric current, and 
therefore would necessitate the existence of magnetic force, if 
we treated the first law of circuitation in its primitive form, 
referred to a stationary medium. Here, however, the motional 
magnetic force, which is ( 44, 49) the vector-product of the 


displacement and the velocity of the medium, comes into play, 
and it is so constructed as to precisely annul all magnetic 
force under the circumstances, and leave the displacement 
(referred to the moving medium) unaffected ; or, in another 
form, it changes the law of circuitation (curl H = J) referred to 
fixed space, so as to refer it in the same form to the moving 

The result is H = 0, and D moves with the medium. 

Similar remarks apply to other stationary states. They are 
unaffected by a common motion of the whole medium and the 
sources (or quasi-sources), and this result is mathematically 
obtained by the motional electric and magnetic forces. 

(2.) Source and Medium in Relative Motion. A Charge 
suddenly jerked into Motion at the Speed of Propaga- 
tion. Generation of a Spherical Electromagnetic Sheet ; 
ultimately Plane. Equations of a Pure Electromagnetic 

54. But the case is entirely altered if the charge and the 
medium have a relative translational motion. 

Start again with charge and medium at rest, and the dis- 
placement stationary and isotropically radial. Next, introduce 
the fact (the truth of which will be fully seen later) that the 
medium transmits all disturbances of the fluxes through 
itself at the speed (MC)~~, which" call -y; and let us suddenly 
set the charge moving in any direction rectilinearly through 
the medium at this same speed, v. Ths question is, what will 
happen ? 

A part of the result can be foreseen without mathematical 
investigation ; the remainder is an example of the theory of the 
simplest spherical wave given by me in "Electromagnetic Waves." 
Let A (in Fig. 1) be the initial position of the charge when it 
first begins to move, and let AC be the direction of its sub- 
sequent motion. Describe a sphere of radius AB = vt ; then, at 
the time t the charge has reached B. Now, from the mere fact 
that the speed of propagation is v, it follows that the dis- 
placement outside the sphere is undisturbed. It is clear that 
there cannot be any change to the right of B, because the 
charge has only just reached that place, and disturbances only 


travel at the same speed as it is moving itself. Similar con- 
siderations applied to the expanding sphere through this 
charge at every moment of its passage from A to B will show 
that no disturbance can have got outside the sphere. The 
radial lines, therefore, represent the actual displacement, as 
well as the original displacement, though of course, in the 
latter case, they extended to the point A. 

We have now to complete the description of the solution. 
There is no displacement whatever inside the sphere BTCDF. 

The displacement emanating from the charge at B, therefore, 
joins on to the external displacement over the spherical surface. 
We can say beforehand that it should do so in the simplest 
conceivable manner, by the shortest paths. On leaving the 
pole B it spreads uniformly in all directions on the surface of 
the sphere, and each portion goes the shortest way to the 
opposite pole D. But it leaks out externally on the way, in 
such a manner that the leakages are equal from equal areas. 
The displacement thus follows the lines of longitude. 


This completes the case so far as the displacement is con- 
cerned. But the spherical surface constitutes an electro- 
magnetic sheet, and corresponding to the displacement there 
is a distribution of coincident, induction. This induction is 
perpendicular to the tangential displacement, and therefore 
follows the lines of latitude. Its direction is up through the 
paper above A (at E, for example), and down through the 
paper below A (at F, for example). The tangential displace- 
ment and induction surface-densities (or fluxes per unit area of 
the sheet), say, D and B , are connected by the equation 

or, = cv . 

Or, if E and H be the equivalent forces got by dividing by 
c and by ft respectively, then, since 2 = 1, 

E = /xvH . 
Or, expressing the mutual directions as well, 

E = VB v; 

where v is the vector velocity of the electromagnetic sheet at 
the place considered. These last are, in fact, as we shall see 
later, the general equations of a wave-front or of a free wave, 
which though it may attenuate as it travels, does not suffer 
distortion by mixing up with other disturbances. 

Now, as time goes on, the charge at B moves off to the right, 
the electromagnetic sheet simultaneously expanding. The ex- 
ternal displacement, therefore, becomes infinitesimal ; likewise 
that on the D side of the shere. Practically, therefore, we 
are finally left with a plane electromagnetic sheet moving 
perpendicularly to itself at speed v, at one point of which 
is the moving charge, from which the displacement diverges 
uniformly in the sheet, following, therefore, the law of the 
inverse first power (instead of the original inverse square), ac- 
companied by a distribution of induction in circles round the 
axis of motion, varying in density with the distance according 
to the same law, and connected with the displacement by the 



above equations. In the diagram, AB has to be very great, 
and the plane sheet is the portion of the spherical sheet round 
B, which is then of insensible curvature. 

(3.) Sudden Stoppage of Charge. Plane Sheet moves on. 
Spherical Sheet generated. Final Result, the Stationary 

55. Having thus turned the radial isotropic displacement 
of the stationary charge into a travelling plane distribution, 
let us suddenly reduce the charge to rest. We know that 

FIG. 2. 

after some time has elapsed, the former isotropic distribution 
will be reassumed ; and now the question is, how will this take 
place ? 

Let B in Fig. 2 be the position of the charge at the moment 
of stoppage and after. Describe a sphere of radius vt, with B 
for centre ; then the point C is where the charge would have 
got at time t after the stoppage had it not been stopped, and 
the plane DOE would have been the position of the plane 


electromagnetic sneer. Now, the actual state of things is- 
described by saying that : 

(1.) The plane sheet DOE moves on quite unaltered, 
except at its core C, where the charge has been taken out. 

(2.) The stationary radial displacement of the charge in its 
new position at B is fully established within the sphere, with- 
out any induction. 

(3.) The internal displacement joins itself on to the external 
in the plane sheet, over the spherical surface, by leaking into 
it and then following the shortest route to the pole C. That is, 
the tangential displacement follows the lines of longitude. 

(4.) The induction in the spherical sheet is oppositely 
directed to before, still, however, following the lines of latitude, 
and being connected with the tangential displacement by the 
former relations. 

In time, therefore, the plane sheet and the spherical sheet go 
out to infinity, and there is left behind simply the radial dis- 
placement of the stationary charge. 

(4.) Medium moved instead of Charge. Or both moved 
with same Relative Velocity. 

56. Now, return to the case of 54, and referring to Fig. 1, 
suppose it to be the charge that is kept at rest, whilst the 
medium is made to move bodily past it from right to left at 
speed v, so that the relative motion is the same as before. We 
must now suppose B to be at rest, the charge being there origi- 
nally, and remaining there, whilst it is A that is travelling 
from right to left, and the spherical surface has a motion com- 
pounded of expansion from the centre A and translation with 
it. Attending to this, the former description applies exactly. 

The external displacement is continuously altering, and there 
is electric current to correspond, but there is no magnetic 
force (except in the spherical sheet), and this, is, as before said, 
accounted for by the motional magnetic force. 

The final result is now a stationary plane electromagnetic 
sheet, as, in fact, described before in 45, where we considered 
the displacement and induction in the sheet to be kept up 
steadily by electric and magnetic forces impressed by the 


Now stop the motion of the medium, without altering 
the position of the charge, and Fig. 2 will show the growth of 
the radial stationary displacement, as in 55, as it is in fact the 
same case precisely after the first moment. 

We can in a similar manner treat the cases of motion and 
stoppage of both charge and medium, provided the relative 
speed be always the speed v t however different from this may 
be the actual speeds. 

(5.) Meeting of a Pair of Plane Sheets with Point-Sources. 
Cancelment of Charges ; or else passage through one 
another; different results. Spherical Sheet with two 
Plane Sheet Appendages. 

57. From the two solutions of 54, 55 (either of which 
may be derived from the other) we may deduce a number of 
other interesting cases. 

Thus, let initially a pair of equal opposite charges +p and 
- p be moving towards one another, each at speed v through 
the medium (which for simplicity we may consider stationary), 
each with its accompanying plane electromagnetic sheet. 
When the charges meet the two sheets coincide, the two dis- 
placements cancel, leaving none, and the two inductions add, 
doubling the induction. We have thus, momentarily, a mere 
sheet of induction. 

Now, if we can carry the charges through one another, with- 
out change in their motion, the two sheets will immediately 
reappear and separate. That is, the plane waves will pass 
through one another, as well as the charges. 

But if the charges cancel one another continuously after their 
first union, a fresh case arises. It is, given a certain plane sheet 
of induction initially, what becomes of it, on the understand- 
ing that there is to be no electrification ? 

The answer is, that the induction sheet immediately splits 
into two plane electromagnetic sheets, joined by a spherical 
sheet, as in Fig. 3. For it is the same as the problem of stop- 
page in 55 with another equal charge of opposite kind moving 
the other way and stopped simultaneously, so that there is no 
electrification ever after. Touching the sphere at the point F 
in Fig. 2 is to be placed the additional plane wave, and the 



CH. II. 

internal displacement is to be abolished. That is to say, in 
Fig. 3 the displacement converges uniformly to F in the plane 
sheet there, then flows without leakage to the opposite pole C 
along the lines of longitude, and there diverges uniformly in 
the other plane sheet. Each displacement sheet has its corre- 
sponding coincident induction, according to the former formulae. 
They all move out to infinity, leaving nothing behind, as there 
is no source left. 

Fio. 6. 

(6.) Spherical Sheet without Plane Appendages produced by 
sudden jerking apart of opposite Charges. 

58. Similarly, let there be a pair of coincident or infinitely 
close opposite charges, with no displacement, and let them be 
suddenly jerked apart, each moving at the speed of propaga- 
tion of disturbances. The result is simply a single spherical 
wave, without plane appendages, and without leakage of the 
displacement. The charges are at opposite poles, at the ends 
of the axis of motion, and the displacement just flows over the 


surface from one to the other symmetrically. There is the 
usual induction B = /wD to match. 

Fig. 3 also shows this case, if we leave out the plane sheets and 
suppose the positive charge to be at F and the negative at C. 

After a sufficient time, we have practically two widely sepa- 
rated plane electromagnetic sheets, although they are really 
portions of a large spherical sheet. 

Now, imagine the motion of the two charges to be reversed ;. 
if we simultaneously reverse the induction in the spherical 
sheet, without altering the displacement, it will still be an 
electromagnetic sheet, but will contract instead of expanding. 
It will go on contracting to nothing when the charges meet. 
If they are then stopped nothing more happens. But if the 
charges can separate again, the result is an expanding spherical 
electromagnetic sheet as before. 

(7.) Collision of Equal Charges of same Name. 

59. If, in the case of colliding plane sheets with charges, 
57, they be of the same name, then, on meeting, it is the- 
induction that vanishes, whilst the displacement is doubled. 
That is, we have momentarily a plane sheet of displacement. 

If the charges be kept together thereafter, this plane 
sheet splits into two plane electromagnetic sheets joined by 
a spherical sheet. At the centre of the last is the (doubled) 
charge 2/5, which sends its displacement isotropically to the 
surface of the sphere, where it is picked up and turned round 
towards one pole or the other. The equator of the sphere is the 
line of division of the oppositely flowing displacements. The 
displacement gets greater and greater as the poles are neared, 
the total amount reaching each pole being p (half the central 
charge), which then diverges in the plane sheet touching the pole. 

The final result, when the waves have gone out to infinity,. 
is, of course, merely the stationary field of the charge 2p. 

(8.) Hemispherical Sheet. Plane, Conical and Cylindrical 

60. If a charge be initially in contact with a perfectly 
conducting plane, and be then suddenly jerked away from it at 
the speed v, the result is merely a hemispherical electromag- 
netic shell. The negative charge, corresponding to the moving 



CH. II. 

point-charge, expands in a circular ring upon the conducting 
plane, this ring being the equator of the (complete) sphere. 

This case, in fact, merely amounts to taking one-half of the 
solution in 58, and then terminating the displacement 
normally upon a conductor. 

In Fig. 4, A is the original position of p on the conducting 
plane CAE, and when the charge has reached B the displace- 
ment terminates upon the plane in the circle DF. 

FIG. 4. 

Instead of a plane conducting boundary, we may similarly 
have conical boundaries, internal and external (or one conical 
boundary alone), with portions of perfect spherical waves run- 
ning along them at the speed v. 

If the two conical boundaries have nearly the same angle, 
and this angle be small, we have a sort of concentric cable 
(inner and outer conductor with dielectric between), of con- 
tinuously increasing thickness. The case of uniform thickness 
is included as an extreme case ; the (portion of the) spherical 
wave then becomes a plane wave. 


General Nature of Electrified Spherical Electromagnetic 

61. The nature of a spherical electromagnetic sheet 
expanding or contracting at speed v, when it is charged in 
an arbitrary manner, may also be readily seen by the foregoing. 
So far, when there has been electrification on the sheet, it has 
been a solitary point-charge or else a pair, at opposite poles. 
Now, the general case of an arbitrary distribution of electrifi- 
cation can be followed up from the case of a pair of 
charges not at opposite ends of a diameter, and each 
of these may be taken by itself by means of an opposite 
charge at the centre or externally, so that integration docs 
the rest of the work. When there is a pair of equal charges 
of opposite sign we do not need any external or internal 
complementary electrification, but we may make use of them 
argumentatively ; or we may let one charge leak outward, 
the other inward, and have the external and internal elec- 
trification in reality. The leakage should be of the isotropic 
character always. But the internal electrification need not be 
at the central point. It may be uniformly distributed upon a 
concentric sphere. This, again, may be stationary, or it may 
itself be in motion, expanding or contracting at the speed v. 
The external electrification, too, may be on a concentric 
spherical surface, which may be in similar motion. The 
sheets, too, may be of finite depth, and arbitrarily electrified, 
so that we have any volume-distribution of electrification 
moving in space in radial lines to or from a centre, accom- 
panied by electromagnetic disturbances arranged in spherical 

There is thus a great variety of ways of making up problems 
of this character, the nature of whose solutions can be readily 
pictured mentally. 

Two charges, q l and q^ for example, on a spherical sheet. One 
way is to put - q t and q 2 at the centre, and superpose the 
two solutions. Or the charges - (q l and q 2 ) may be put exter- 
nally, with isotropic leakage. Or part may be inside, and part 
outside. Or we may have no complementary electrification at 
all, but lead the displacement away into plane waves touching 
the sphere. 


Only when the total charge on the spherical sheet is zero can 
we dispense with these external aids (which to use depending 
upon the conditions of the problem) ; then the displacement 
has sources and sinks on the sheet which balance one another. 
The corresponding induction is always perpendicular to the 
resultant tangential displacement, and is given by the above 
formula, or B = /*VvD , where D is the tangential displace- 
ment in the sheet (volume density x depth). 

One case we may notice. If the density of electrification be 
uniform over the surface, there is no induction at all. That is, 
it is not an electromagnetic sheet, but only a sheet of electrifi- 
cation, without tangential displacement, and therefore without 

So, with a condenser consisting of a pair of concentric shells 
uniformly electrified, either or both may expand or contract 
without magnetic force. This is, however, not peculiar to the 
case of motion at speed v. Any speed will do. But in general, 
if the speed be not exactly v, there result diffused disturb- 
ances. The electromagnetic waves are no longer of the same 
pure type. 

General Remarks of the Circuital Laws. Ampere's Rule 
for deriving the Magnetic Force from the Current. 
Rational Current-element. 

62. The two laws of circulation did not start into full 
activity all at once. On the contrary, although they express 
the fundamental electromagnetic principles concerned in the 
most concise and clear manner, it was comparatively late in 
the history of electromagnetism that they became clearly re- 
cognised and explicitly formularised. We have not here, how- 
ever, to do the work of the electrical Todhunter, but only to 
notice a few points of interest. 

The first law had its first beginnings in the discovery of 
Oersted that the electric conflict acted in a revolving manner, 
and in the almost simultaneous remarkable investigations of 
Ampere. It did not, however, receive the above used form of 
expression. In fact, in the long series of investigations in 
electro-dynamics to which Oersted's discovery, and the work of 
Ampere, Henry, and Faraday, gave rise, it was customary to 
consider an element of a conduction current as generating a 


certain field of magnetic force. Natural as this course may 
have seemed, it was an unfortunate one, for it left the question 
of the closure of the current open ; and it is quite easy to see 
now that this alone constituted a great hindrance to progress. 
But so far as closed currents are concerned, in a medium of 
uniform inductivity, this way of regarding the relation between 
current and magnetic force gives equivalent results to those 
obtained from the first law of circuitation in the limited form 
suitable to the circumstances stated. 

If C is the density of conduction current at any place, the 
corresponding field of magnetic force is given by 

at distance r from the current-element 0, if x l be a unit vector 
along r from the element to the point where H is reckoned. 
The intensity of H thus follows the law of the inverse square 
of the distance along any radius vector proceeding from the 
current-element; but, in passing from one radius vector to 
another, we have to consider its inclination to the axis of the 
current by means of the factor sin (where is the angle 
between r and the axis), involved in the vector product. Also, 
H is perpendicular to the plane containing r and the axis of 
the current-element, or the lines of H are circles about this axis. 
But from the Maxwellian point of view this field of H is that 
corresponding to a certain circuital distribution of electric 
current, of which the current-element mentioned is only a 
part ; this complete current being related to the current- 
element in the same way as the induction of an elementary 
magnet is to the intensity of magnetisation of the latter. 
Calling the complete system of electric current a rational 
current-element, it may be easily seen that in a circuital distri- 
bution of rational current-elements the external portion of the 
current disappears by mutual cancelling, and there is left only 
the circuital current made up of the elements in the older 
sense. We may, therefore, employ the formula (1) to calculate 
without ambiguity the magnetic force of any circuital distri- 
bution of current. This applies not merely to conduction 
current (which was all that the older electricians reckoned), 
but to electric current in the wider sense introduced by 



Maxwell. But the result will not be the real magnetic force 
unless the 'distribution of inductivity is uniform. When /* 
varies, we may regard the magnetic force of the current thus 
obtained as an impressed magnetic force, and then calculate 
what induction it sets up in the field of varying inductivity. 
This may be regarded as an independent problem. 

Tn passing, we may remark that we can mount from magnetic 
current to electric force by a formula precisely similar to (1), 
but subject to similar reservations. 

The Cardinal Feature of Maxwell's System. Advice to 

63. But this method of mounting from current to magnetic 
force (or equivalent methods employing potentials) is quite 
unsuitable to the treatment of electromagnetic waves, and is 
then usually of a quite unpractical nature. Besides that, the 
function " electric current " is then often a quite subsidiary 
and unimportant quantity. It is the two fluxes, induction and 
displacement (or equivalently the two forces to correspond), 
that are important and significant; and if we wish to know the 
electric current (which may be quite a useless piece of informa- 
tion) we may derive it readily from the magnetic force by 
differentiation ; the simplicity of the process being in striking 
contrast to that of the integrations by which we may mount 
from current to magnetic force. 

To exemplify, consider the illustrations of plane and 
spherical electromagnetic waves of the simplest type given in 
53 to 61, and observe that whilst the results are rationally 
and simply describable in terms of the fluxes or forces, yet to 
describe in terms of electric current (and derive the rest from 
it) would introduce such complications and obscurities as would 
tend to anything but intelligibility. 

Now, Maxwell made the first law of circuitation (not, how- 
ever, in its complete form) practically the definition of electric 
current. This involves very important and far-reaching con- 
sequences. That it makes the electric current always circuital 
at once does away with a host of indeterminate and highly 
speculative problems relating to supposititious unclosed cur- 
rents. It also necessitates the existence of electric current in 
perfect non-conductors or insulators. This has always been a 


stumbling-block to practicians who think themselves practical. 
But Maxwell's innovation was really the most practical im- 
provement in electrical theory conceivable. The electric 
current in a nonconductor was the very thing wanted to co- 
ordinate electrostatics and electrokinetics, and consistently 
harmonise the equations of electromagnetism. It is the 
cardinal feature of Maxwell's system, and, when properly 
followed up, makes the insulating medium the true medium in 
the transmission of disturbances, and explains a multitude of 
phenomena that are inconceivable in any other theory (unless 
it be one of the same type). But let the theoretical recom- 
mendations (apart from modern experiments), which can only 
be appreciated after a pretty close study of theory, Maxwellian 
and otherwise, stand aside, and only let the tree be judged by 
its fruit. Is it not singular that there should be found people, 
the authors of works on Electricity, who are so intensely pre- 
judiced against the Maxwellian view, which it would be quite 
natural not to appreciate from the theoretical standpoint, as 
to be apparently quite unable to recognise that the fruit has 
any good flavour or savour, but think it no better than Dead 
Sea fruit ? The subject is quite sufficiently difficult to render 
understandable popularly, without the unnecessary obstruc- 
tion evidenced by a carping and unreceptive spirit. The 
labours of many may be required before a satisfactory elemen- 
tary presentation of the theory can be given. So much the 
more need, therefore, is there for the popular writer to recognise 
the profound significance of the remarkable experimental work 
of late years, a significance he appears to have so sadly missed. 
When that is done, then will be the time for an understanding 
of Maxwell's views. Let him have patience, and believe that 
it is not all speculative metaphysics because it is not to his 
present taste. Never mind the ether disturbances playing 
pranks with the planets. They can take care of themselves. 

Changes in the Form of the First Circuital Law. 

64. Two or three changes I have made* in Maxwell's form 
of the first circuital law. One is of a formal character, the 
introduction of the h term to express the intrinsic force of 

* "Electromagnetic Induction and its Propagation," The Electrician^ 
1885, January 3, and later : or reprint. 

F 2 


magnetisation. This somewhat simplifies the mathematics, 
and places the essential relations more clearly before the eye. 
Connected with this is a different reckoning of the energy of an 
intrinsic magnet, in order to get consistent results. 

The second change, which is not merely one of a formal 
character, but is an extension of an obligatory character, is the 
introduction of the h term to represent the motional magnetic 
force. In general problems relating to electromagnetic wavea 
it is equally important with the motional electric force. 

The third change is the introduction (first done, I think, by 
Prof. Fitzgerald) of the term to explicitly represent the con- 
vection-current or electrification in motion as a part of the true 
current. Although Maxwell did not himself explicitly repre- 
sent this, which was a remarkable oversight, he was strongly 
insistent upon the circuital nature of electric current, and 
would doubtless have seen the oversight the moment it was 
suggested to him. Now, there are spots on the sun, and I see 
no good reason why the many faults in Maxwell's treatise 
should be ignored. Tt is most objectionable to stereotype the 
work of a great man, apparently merely because it was so great 
an advance, and because of the great respect thereby induced. 
The remark applies generally ; to the science of Quaternions, 
for instance, which, if I understand rightly, Prof. Tait would: 
preserve in the form given to it by Hamilton. In application 
to Maxwell's theory, I am sure that it is in a measure to the 
recognition of the faults in his treatise that a clearer view of. 
the theory in its broader sense is due. 

Introduction of the Second Circuital Law. 

65. The second circuital law, like the first, had an experi- 
mental origin, of course, and, like the first, was long in approxi- 
mating to its present form much longer, in fact, though in a 
different manner. The experimental foundation was Faraday's 
recognition that the voltage induced in a conducting circuit 
was conditioned by the variation of the number of lines of force 
through it. 

But, rather remarkably, mathematicians did not put this 
straight into symbols for an elementary circuit, but went to 
work in a more roundabout way, and expressed it through the 
medium of an integration extended along a concrete circuit > 


or else, of an equation of electromotive force containing a 
function called the vector potential of the current, and another 
potential, the electrostatic, working together not altogether in 
the most harmoniously intelligible manner in plain English, 
muddling one another. It is, I believe, a fact which has been 
recognised that not even Maxwell himself quite understood how 
they operated in his " general equations of propagation." We 
need not wonder, then, that Maxwell's followers have not found 
it a very easy task to understand what his theory really meant, 
and how to work it out. I had occasion to remark, some years 
since, that it was very much Maxwell's own fault that his 
views obtained such slow acceptance; and, in now repeating the 
remark, do not abate one jot of my appreciation of his work, 
which increases daily. For he devoted the greater part of his 
treatise to the working out and presentation of results which 
could be equally well done in terms of other theories, and gave 
only a very cursory and incomplete exposition of what were 
peculiarly his own views and their consequences, which are 
of the utmost importance. At the same time, it is easily to be 
recognised that he was himself fully aware of their importance, 
by the tone of quiet confidence in which he wrote concerning 

Finding these equations of propagation containing the two 
potentials unmanageable, and also not sufficiently comprehen- 
sive, I was obliged to dispense with them ; and, going back to 
first principles, introduced* what I term the second circuital law 
as a fundamental equation, the natural companion to the first. 
The change is, I believe, a practical one, and enables us to con- 
siderably simplify and clarify the treatment of general ques- 
tions, whilst bringing to light interesting relations which were 
formerly hidden from view by the intervention of the vector 
potential A, and its parasites J and "VP. 

Another rather curious point relates to the old German 
electro-dynamic investigations and their extensions to endeavour 
to include, supersede, or generalise Maxwell by anti-Maxwellian 
methods. It would be slaying the slain to attack them; but 
one point about them deserves notice. The main causes of the 
variety of formulae, and the great complexity of the investiga- 
tions, were first, the indefiniteness produced by the want of 
* See footnote, p. 67. 



CH. II. 

circuitality in the current, and next the potential methods 
employed. [J. J. Thomson's " Report on Electrical Theories " 
contains an account of many. As a very full example, that 
most astoundingly complex investigation of Clausius, in the 
second volume of his " Mechanische Warmetheorie," may be 
referred to.] But if the critical reader will look through these 
investigations, and eliminate the potentials, he will find, as a 
useful residuum, the second circuital law; and, bringing it 
into .full view, will see that many of these investigations are 
purely artificial elaborations, devoid of physical significance; 
gropings after mares' nests, so to speak. Now, using this law 
in the investigations, it will be seen to involve, merely as a 
matter of the mathematical fitness of things, the use of another, 
viz., the first circuital law, and so to justify Maxwell in his 
doctrine of the circuitality of the current. The useful moral 
to be deduced is, I think, that in the choice of variables to ex- 
press physical phenomena, one should keep as close as possible 
to those with which we are experimentally acquainted, and 
which are of dynamical significance, and be on one's guard 
against being led away from the straight and narrow path in 
the pursuit of the Will-o'-the-wisp. 

As regards the terms in the expression of the second law 
which stand for unknown properties, the} 7 may be regarded 
merely as mathematical extensions which, by symmetrizing the 
equations, render the correct electric and magnetic analogies 
plainer, and sometimes assist working out. But some other 
extensions of meaning, yet to be considered, have a more sub- 
stantial foundation. 

Meaning of True Current. Criterion. 

66. One of these extensions refers to the meaning to be 
attached to the term current, electric or magnetic respectively. 
As we have seen, electric current, which was originally conduc- 
tion current only, had a second part, the displacement current, 
added to it by Maxwell, to produce a circuital flux; and further, 
when there is electrification in motion, we must, working to the 
same end, add a third term, the convection current, to preserve 

A fourth term may now be added to make up the " true " 
current, when the medium supporting the fluxes is in motion. 


Thus, let us separate the motional electric and magnetic forces 
from all other intrinsic forces that go with them in the two 
circuital laws (voltaic, thermo-electric, <feq., 39) ; denoting the 
former by e and h, and the latter by e and h . The two 
equations of circuitation [(12), (13) 38, and (6), (7) 52)] now 

curl(H-h -h) = J = C + D + u/>, . . . (1) 

- curl (E - e - e) = G = K + B + wcr. . . (2) 
Now transfer the e and h terms to the right side, producing 

curl (H-h ) = J =0 + D + u/o + j,. . . (3) 

- curl (E - e c ) = G = K + B + wo- + g, . . (4) 
where j and g are the auxiliaries before used, given by 

j = curl h, g = - curl e. 

It is the new vectors J and G which should be regarded as 
the true currents when the medium moves. 

As the auxiliaries j and g are themselves circuital vectors, 
there may not at first sight appear to be any reason for further 
complicating the meaning of true current when separated into 
component parts, for the current J is circuital, and so is J , 
which is of course the same as J when the medium is station- 
ary. The extension would appear to be an unnecessary one, of 
a merely formal character. 

On the other hand, it is to be observed that from the 
Maxwellian method of regarding the current as a function of 
the magnetic force, the extended meaning of true current is not 
a further complication, but is a simplification. For whereas 
in the equation (2) we deduct from the force E of the flux 
not only the intrinsic force e but also the motional force e 
to obtain the effective force whose curl measures the current ; 
on the other hand, in (4) we deduct only the intrinsic force. 
Away, therefore, from the sources of energy which are inde- 
pendent of the motion of the medium, the force whose curl is 
taken in (4) is the force of the flux, which specifies the elec- 
tric state of the medium, whether it be stationary or moving. 

To show the effect of the change, consider the example of 
56, in which the medium moves past the charge, and there is 
continuously changing displacement outside a certain sphere. 
If we consider J the true current, we should say there is elec- 


trie current, although there is no magnetic force. But accord- 
ing to the other way there is no true current, and the absence 
of magnetic force implies the absence of true electric current. 

But still this question remains, so far, somewhat of a con- 
ventional one. Is there any test to be applied which shall 
effectually discriminate between J and J as the true current ? 
There would appear to be one and only one ; viz., that e being 
an intrinsic force, its activity, whatever it be, say e x, expresses 
the rate of communication of energy to the electromagnetic 
system from a source not included therein, and not connected 
/ with the motional force e. When the medium is stationary x 
is J, and the question is, is x to be J or J (or anything else) 
when the medium moves ? 

Now this question can only be answered by making it a part 
of a much larger and more important one ; that is to say, by a 
comprehensive examination of all the fluxes of energy concerned 
in the equations (1), (2), or (3), (4), and their mutual harmo- 
nisation. This being done, the result is that e J is the activity 
of e , so that it is J that is the measure of the true current, 
with the simpler relation to the force of the flux; and it is 
naturally suggested that in case of further possible extensions, 
we should follow in the same track, and consider the true cur- 
rent to be always the curl of (H - h ), independently of the 
make up of its component parts. 

This is not the place for a full investigation of this complex 
question, but the main steps can be given ; not for the exhibi- 
tion of the mathematical working, which may either be taken 
for granted, or filled in by those who can do it, but especially 
with a view to the understanding in a broad manner of the 
course of the argument. Up to the present I have used the 
notation of vectors for the concise and plain presentation of 
principles and results, but not for working purposes. This 
course will be continued now. How to work vectors may form 
the subject of a future chapter. It is not so hard when you 
know how to do it. 

The Persistence of Energy. Continuity in Time and Space 
and Flux of Energy. 

67. The principle of the conservation or persistence of 
energy is certainly as old as Newton, when viewed from the 


standpoint of theoretical dynamics. But only (roughly speak- 
ing) about the middle of the present century did it become 
recognised by scientific men as a universal truth, extending to 
all the phenomena of Nature. This arose from the experi- 
mental demonstrations given (principally those of Joule), that 
in various cases in which energy disappeared from one form, 
it was not lost, but made its appearance in other forms, and 
to the same amount. 

The principle is now-a-days a sure article of scientific faith. 
Its ultimate basis is probably the conviction (a thoroughly 
reasonable one) that the laws of motion hold good in the in- 
visible world as well as in the visible, confirmed by the repeated 
experimental verification. But the principle is now believed in 
quite generally ; by the ordinary unscientific man, for example, 
who could not tell you correctly what the principle meant, or 
energy either, to save his life, although he might tell you that 
he could not conceive the possibility of energy being destroyed. 
He goes by faith, having taken it in when young. 

So it will be with the modern view of the ether as the 
medium through which energy is being sent when a wire sup- 
ports a current. Only train up the young to believe this, and 
they will afterwards look upon the notion of its going through 
the conductor as perfectly absurd, and will wonder how anyone 
ever could have believed it. But before this inevitable state of 
things comes to pass, an intermediate period must be passed 
through, in which scientific men themselves are learning to 
believe the modern view thoroughly from the evidence of its 
truth, and to teach it to others. This period is certainly not 
yet come to its end ; for almost weekly we may read about 
" the velocity of electricity in wires," such language emanating 
from scientific men who are engaged in repeating and extend 
ing Hertz's experiments. 

The principle of the continuity of energy is a special form of 
that of its conservation. In the ordinary understanding of the 
conservation principle it is the integral amount of energy that 
is conserved, and nothing is said about its distribution or its 
motion. This involves continuity of existence in time, but not 
necessarily in space also. 

But if we can localise energy definitely in space, then we are 
bound to ask how energy gets from place to place. If it 


possessed continuity in time only, it might go out of existence 
at one place and come into existence simultaneously at 
another. This is sufficient for its conservation. This view, 
however, does not recommend itself. The alternative is to 
assert continuity of existence in space also, and to enunciate 
the principle thus : 

When energy goes from place to place, it traverses the 
intermediate space. 

This is so intelligible and practical a form of the principle, 
that we should do our utmost to carry it out. 

The idea that energy has position, therefore, naturally 
involves the idea of a fiux of energy. Let A be the vector flux 
of energy, or the amount transferred per unit of time per unit 
of area normal to the direction of its transfer ; and let T be 
the density of the energy. We may then write 

conv A = f, (5) 

or divA=-T (6) 

That is, the convergence of the flux of energy is accounted for 
by increased energy in the unit volume where it converges ; 
or, the divergence of the flux of energy, or the rate at which 
it leaves the unit volume, is accompanied by simultaneous and 
corresponding decrease in the density of the energy. 

Examples. Convection of Energy and Flux of Energy due 
to an active Stress. Gravitational difficulty. 

68. Now, there are numerous cases in which the flux of 
energy is perfectly plain, and only needs to be pointed out to- 
be recognised. The simplest of all is the mere convection of 
energy by motion of the matter with which it is associated. A 
body in translational motion, for instance, carries its energy 
with it; not merely the kinetic energy of its translational 
motion, but also its rotational energy if it- rotates, and like- 
wise the energy of any internal vibratory or rotatory motions 
it may possess, and other energy not known to be kinetic, 
and therefore included under "potential energy," which we 
may have good reason to localise in the body. Even 
if part of the energy be outside the body, yet, if it be 
associated with the body, it will travel with it, on the whole, 


and so (at least sometimes) may be considered to be rigidly 
attached to it. This case may be regarded as the convection 
of energy, if we do not go too closely into detail. 

When energy is thus conveyed, the energy flux is qT, where 
q is the velocity and T the density of the energy conveyed (of 
any kind). If, then, all energy were conveyed, equation (5) 
would become 

convaT = t (7) 

Now this equation, if we regard T as the density of matter, is 
the well known equation of continuity of matter used in hydro- 
dynamics and elsewhere. 

This brings us to Prof. Lodge's theory of the identity of 
energy. (Phil. Mag., 1885.) Has energy personal identity, 
like matter ? I cannot see it, for one ; and think it is pushing 
the principle of continuity of energy, which Prof. Lodge was 
writing about, too far. It is difficult to endow energy with 
objectivity, or thinginess, or personal identity, like matter. The 
relativity of motion seems to be entirely against the idea. Nor 
are we able to write the equation (7) in all cases. Energy may 
be transferred in other ways than by convection of associated 
matter. If we atomise the energy we can then imagine the q. 
above to be the velocity of the energy, not of the matter. But 
as the science of dynamics is at present understood, we cannot 
make use of this idea profitably, I think. 

The chief reason why the founders of the modern science of 
energy did not explicitly make use of the idea of the continuity 
of energy was probably the very obscure nature of gravitational 
energy. Where was it before it became localised as the kinetic 
energy of a mass ? It is of no use to call it potential energy 
if that is to explain anything, which it does not. It would, seem 
that the energy must have been in the ether, somewhere, and 
was transferred into the body, somehow. This makes the ether 
the great store-house of all gravitational energy. But we are 
entirely ignorant of its distribution in the ether, and of its 
mode of transference. 

Observe here that ether must be regarded as a form of 
matter, because it is the recipient of energy, and that is the 
characteristic of ordinary matter. There is an unceasing- 
enormous flux of energy through the ether from the Sun, for 


instance, and we know that it takes several minutes to come : 
it comes through the ether, without bringing the ether with it; 
it is not a convection of energy, therefore. Of course, to avoid 
confusion, it is well to distinguish ether from ordinary matter 
by its separate name ; but it is important to note that it has 
some of the characteristics of ordinary matter. It need not be 
gravitating matter ; it is, perhaps, more likely to be the medium 
of gravitational action than to gravitate itself. 

In default of ability to represent the flux of gravitational 
energy its entry into a body must be otherwise represented than 
by the convergence of a flux. Turn (7) into 

f<l + convA = f (8) 

Here fq. is the activity of the force f, and indicates energy com- 
municated to the unit volume in an unstated manner, generated 
within the unit volume, so to speak. The force f is thus an 
impressed force. By this device we can allow for unknown 
fluxes of energy. We should also explicitly represent the 
convective flux. Thus, 

fq + conv(qT + A) = f, .... (9) 

where A is the flux of energy other than convective, not asso- 
ciated with f. 

When a stress works, for example, we shall have a flux of 
energy, A, which can hardly be considered to be convective, in 
the sense explained, a flux of energy through matter or ether, 
as in wave propagation through an elastic solid, or radiation 
through the ether. We can, however, sometimes reduce it to 
the form qT 1? and then it would appear to be convection of 

This occurs in a moving frictionless fluid, for example, when 
the stress is an isotropic pressure p. The flux of energy other 
than convective is jpq. But when we pass to a less ideal case, 
as an elastic solid, in which the stress is of a more general 
character, the energy flux expressive of the activity of the 
stress takes the form 

A=-P fl (z, ...... (10) 

where P 7 is the vector expressing the pull per unit area on the 
plane perpendicular to the velocity q. In this equation q is 
the tensor of q. 


Specialised form of expression of the Continuity of Energy. 

69. Passing to more practical and specialised forms of 
expression of the continuity of energy, it is to be observed 
first that we are not usually concerned with a resultant flux 
of energy from all causes, but only with the particular forms- 
relating to the dynamical question that may be under con- 
sideration. Secondly, that it is convenient to divide the 
energy above denoted by T into different parts, denoting 
potential energy, and kinetic energy, and wasted energy. 
Thus we have the following as a practical form, 

fq + conv[A + q(U + T)] = Q + U + T, . (11) 

where the terms in the square brackets indicate the energy 
flux, partly convective, with the factor q, U being the density 
of potential energy, and T that of kinetic energy, and partly 
non-convective, viz., A, which may be due to a working stress. 
Also fq is the activity of impressed force (here merely a trans- 
lational force). Thus on the left side of equation (11) we- 
have a statement of the supply of energy to the unit volume- 
fixed in space. On the right side we account for it by the rates- 
of increase of the stored potential and kinetic energy, and by 
Q, which means the rate of waste of energy in the unit volume. 
The wasted energy is also stored, at least temporarily ; but not 
recoverably, so that we may ignore energy altogether after it is 
once wasted. 

In place of the term fq, in which the impressed force is 
translational, we may have other terms possessing a similar 
meaning, indicating a supply of energy from certain sources. 
These sources, too, may be not external, but internal or intrin- 
sic, as for instance when there is a thermo-electric or voltaic- 
source of energy within the unit volume considered. 

Also, the terms A, U, T, and Q may have to be split up 
into different parts, according to the nature of the dynamical 
connections. The important thing to be grasped is, that when- 
ever we definitely localise energy we can obtain an equation 
showing its continuity in space and time, and that when we 
can only partially localise it, we can still, by proper devices, 
allow for the absence of definiteness. The equation is simply 
the equation of activity of the dynamical system, suitably 



CH. II. 

arranged to show the flux of energy, and can always be obtained 
when the equations of motion are known and also the nature of 
the stored and wasted energies. 

Electromagnetic Application. Medium at Eest. The 
Poynting Flux. 

70. Now, in the electromagnetic case, the "equations of 
motion " are the two circuital laws, and to form the equation 
of activity, we may multiply (1) by (H - h - h) and (2) by 
(E - e - e) and add the results ; or else multiply (3) by (H - h ) 
and (4) by (E - e ) and add the results. The equation of activity 
thus obtained has then to be dynamically interpreted in accord- 
ance with the principle of continuity of energy. 

When the medium is stationary, there is no difficulty with 
the interpretation. We obtain 

e J + h G = Q + U + T + divW, . . (12) 
where W is a new vector given by 

W = V(E-e )(H-h ), . . . (13) 

and U is the electric energy, T the magnetic energy, and Q the 
waste per unit volume. 

On the left side of (12) is exhibited the rate of supply of 
energy from intrinsic sources. On the right side it is accounted 
for partly by the waste Q, and by increase of the stored energy 
U and T. The rest is exhibited as the divergence of the flux W 
given by (13). We conclude, therefore, that W expresses the 
flux of energy in the electromagnetic field when it is stationary. 

This remarkable formula was first discovered and interpreted 
by Prof. Poynting [Phil. Trans., 1884, Pt. 2], and independently 
by myself a little later. It was this discovery that brought 
the principle of continuity of energy into prominence. But it 
should be remembered that there is nothing peculiarly electro- 
magnetic about a flux of energy. It is here made distinct, 
because the energy is distinctly localised in Maxwell's theory. 

The flux of energy takes place in the direction perpendicular 
to the plane containing the electric and magnetic forces (of the 
field), say 

W = VE 1 H 1 , (14) 

if E = e + E 1 ; H = h + H r 


The formula is not a working formula, in general, for to 
know W we must first know the distribution of the electric and 
magnetic forces. But it is a valuable and instructive formula 
for all that. Its discovery furnished the first proof that Max- 
well's theory implied that the insulating medium outside a 
conducting wire supporting an electric current was the medium 
through which energy is transferred from its source at a dis- 
tance, provided that we admit the postulated storage of energy. 
For inside a wire the electric force is axial, and the magnetic 
force circular about the axis ; the flux of energy is therefore 
radial. When the current is steady, it comes from the boundary 
of the wire, and ceases at its axis. It delivers up energy on the 
way, which is wasted in the Joule-heating. But when the 
current is not steady, the magnetic and electric energy will be 
also varying. 

If we work out the distributions of electric and magnetic 
force outside the wire, according to the conditions to which it 
is subjected and its environment, we can similarly fully trace 
how the energy is supplied to the wire from its source. It is 
passed out from the source into the dielectric medium, and 
then converges upon the wire where it is wasted. The flux of 
energy usually takes place nearly parallel to the wire (because 
the electric force is nearly perpendicular to its boundary) ; 
its slight slope towards the wire indicates its convergence 
thereupon. If the wire had no resistance, there would be* 
no convergence of energy upon it, the flux of energy 
would be quite parallel to it. Details are best studied 
in the concfete application, and it is only by the con- 
sideration of variable states, and the propagation of electro- 
magnetic waves, that we can obtain a full understanding of 
the meaning of W considered as a flux of energy. 

In the case of a simple progressive plane wave disturbance, 
in which a distribution of E and H (mutually perpendicular) in 
the plane of the wave is propagated unchanged through a 
medium at constant speed, it is a self-evident result that the 
energy of the disturbance travels with it. The flux of energy 
is, therefore (since e = 0, h = 0), 

W = v(U + T) = VEH, ..... (15) 
where v is the velocity of the wave, and U, T are the densities 


(which are here equal) of the electric and magnetic energies. 
In this example, which is approximately that of radiant energy 
from the sun, the idea of a flux of energy, and the conclusion 
that its proper measure is the density of the energy multiplied 
by the wave velocity, are perfectly plain and reasonable. Now, 
many cases of the propagation of waves along wires can be 
reduced to this simple case, with a correction for the resistance 
of the wire, and other cases can be represented by two or more 
oppositely travelling plane disturbances. In a very complex 
electromagnetic field, the flux of energy is necessarily also very 
complex, and hard to follow ; but the fundamental principles 
concerned are the same throughout. 

The flux of energy W arises from the internal structure of 
the ether. It is somewhat analogous to the activity of a stress. 
But the only dynamical analogy that is satisfactory in this 
respect is that furnished by Sir W. Thomson's rotational ether, 
when interpreted in a certain manner, so that 2E shall repre- 
sent a torque, and H the velocity of the medium, with the 
result that (on this understanding) VEH is the flux of energy, 
whilst U is the potential energy of the rotation, and T the 
kinetic translational energy. But it is very difficult to extend 
this analogy to include electromagnetic phenomena more com- 
prehensively. [See Appendix at the end of this chapter.] 

Extension to a Moving Medium. Full interpretation of 
the Equation of Activity and derivation of the Flux 
of Energy. 

71. Passing now to the case of a moving medium, we shall 
obtain, from equations (1), (2), 66, the equation of activity, 

(e + e) J + (h + h)G = E J + HG 

+ div V(B - 0o - e)(H - ho - h). (16) 

Or from equations (3), (4), 66, we may get the form 

e J + h G = E J + HG + div V(E - e )(H - h ), (1 7) 

and from these a variety of other forms may be derived. 
The dynamical interpretation in accordance with the prin- 
ciple of continuity of energy is not so easy as in the former 
case. I have recently given a full discussion of these 


equations in another place. The following gives an outline 
of the results : 

Since there is a flux of energy when the medium is stationary, 
there will still be a flux of this kind (i.e., independent of the 
motion) when it is moving, whatever other flux of energy there 
may then be. I conclude that the Poynting flux W still 
preserves the form 

W = V(E-e )(H-h ) 5 .... (18) 

from the fact that disturbances are propagated through a 
medium endowed with a uniform translational motion in the 
same manner as when it is at rest. Otherwise we could only 
know that it must reduce to this form when at rest. 
Next, there is the convective flux of energy 

...... (19) 

where q. is the velocity of the medium. 

Thirdly, there is a flux of energy representing the activity, 
A, of the electromagnetic stress. It is given by 

. . . (20) 

Fourthly, there is, in association with this stress, a convective 
flux of other energy, say, 

1(U + T ) ....... (21) 

The complete flux of energy is the sum X, of these four 
vectors, i.e. : 

X = W + q(U + T) + A + q(U + T ), . . (22) 

W and A being given by (18) and (20). 

Finally, the activity of the intrinsic forces is 

OoJo + VJo, ..... (23) 

so that J is the true electric current. 

We, therefore, have the equation of activity brought to the 
standard form 

e J + h 0- = (Q + U + T) + (Q + U + T ) + div. X, (24) 

which is a special form of equation (11), with the convergence 
of the energy flux in it replaced by its divergence (the negative 
of convergence) on the other side of the equation. 


But the unknown terms, on the right side of (24), with the 
zero suffix, may be entirely eliminated, including those in X, by 
making use of the secondary equation of translational activity 

T ), . . (25) 

where F is the translational force due to the stress. * 
When this is done, equation (24) takes the form 

e J + h G = Q + U + T + Fq + div.[W + A + q (U + T)] (26) 

where the energy flux represented includes the Poynting flux, 
the stress flux, and the convective flux of electric and magnetic 

We may observe that in the expression (20) for A occurs the 
term - q (U + T), so that this term may be eliminated from (26), 
making the energy flux in it become 

V(B - e )(H - ho) - VeH -, VEh. . . (27) 

But these changes, with a view to the simplification of expres- 
sion, cause us to altogether lose sight of the dynamical signi- 
ficance of the equation of activity, and of the stress function. 
Equation (26) is, therefore, the best form. 

It should be understood that it is an identity, subject to the 
two laws of circuitation and the distribution of energy accord- 
ing to that of and H in the field. But it should be also 
mentioned that in the establishment of (26) it has been assumed 
that the medium, as it moves, carries its intrinsic properties of 
permittivity and inductivity with it unchanged. That is, these 
properties do not alter for the same portion of the medium, 
irrespective of its position, although within the same unit 
volume these properties may be changing by the exit of one 
and entry of another part of the medium of different permit- 
tivity and inductivity ; understanding by " medium" whatever 
is supporting the fluxes, whether matter and ether together, or 
ether alone ; and it is also to be understood that the three 
velocities q, u, and w are identical, or that electrification, 
which is always found associated with matter, moves with the 
medium, which is then the matter and ether, moving together. 
In other respects equation (26) is unrestricted as regards either 
homogeneity and isotropy in respect to permittivity, inductivity, 
and conductivity (electric, and fictitious magnetic). 

[* I regret to have misrepresented Dr. Burton's notion of a moving strain 
figure. He does it entirely by conservative elastic forces, and my objection 
does not apply.] 


Whether, when matter moves, it carries the immediately 
surrounding ether with it, or moves through the ether, or only 
partially carries it forward, and what is the nature of the 
motion produced in the ether by moving matter, are questions 
which cannot be answered at present. Optical evidence is 
difficult of interpretation, and conclusions therefrom are con- 

But in ordinary large scale electromagnetic phenomena, it 
can "make very little difference whether the ether moves or 
stands stock-still in space. For the speed with which it pro- 
pagates disturbances through itself is so enormous that if the 
ether round a magnet were stirred up, artificially, like water 
in a basin, with any not excessive velocity, the distortion in 
the magnetic field produced by the stirring would be next to 

Strictly speaking, when matter is strained its elastic and 
other constants must be somewhat altered by the distortion of 
the matter. The assumption, therefore, that the permittivity 
and inductivity of the same part of the medium remain the 
same as it moves is not strictly correct. The dependence of 
the permittivity and inductivity on the strain can be allowed for 
in the reckoning of the stress function. This matter has been 
lately considered by Prof. Hertz. But the constants may also 
vary in other ways. It is unnecessary to consider here these 
small corrections. As usual in such cases, the magnitude of 
the expressions for the corrections is out of all proportion to 
their importance, in relation to the primary formula to which 
they are added. 

Derivation of the Electromagnetic Stress from the Flux 
of Energy. Division into an Electric and a Magnetic 

72. From the form of the expression for A in equation (20), 
viz., the flux- of energy due to the stress, we may derive the 
expression for the electromagnetic stress itself. If the stress 
were of the irrotational type considered in works on Elasticity, 
we could do this by means of the formula (10), 68. ' But for a 
stress of the most general type the corresponding formula is 




where Q, is the stress vector conjugate to P q ; these are identi- 
cally the same when the stress is irrotational. This gives 

Q N = D.EN + B.HN-N(U + T), . . (29) 

from which we obtain P N by merely exchanging E and D, and 
H and B ; thus 

P N = E.DN + H.BN-N(U + T). . . . (30) 

This is the stress vector for any plane denned by N, a unit 
vector normal to the plane. 

But it is only in eolotropic bodies that we have to distinguish 
between the directions of a force and of the corresponding 
flux. Putting^ these on one side, and considering only ordi- 
nary isotropy, the interpretation is simple enough. It will be 
observed that the electromagnetic stress (30) divides into an 
electric stress 

E.DN-N.JED, (31) 

and a magnetic stress 

H.BN-N.JHB (32) 

To find their meaning, take N in turns parallel to and perpen- 
dicular to the force E (or to D, since its direction is the same). 
In the first case, the stress (31) becomes 

N(ED - JED) = N.JED = NU, 

indicating a tension parallel to the electric force of amount U 
per unit area. 

In the second case, when N is perpendicular to E, we have 
DN = 0, so that the stress is 


that is, a pressure of amount U. This applies to any direction 
perpendicular to E, so that the electric stress consists of a 
tension U parallel to the electric force, combined with an equal 
lateral pressure. 

Similarly the magnetic stress consists of a tension T parallel 
to the magnetic force, combined with an equal lateral pressure. 

It will also be found that the tensor of the electric stress 
vector is always U, and that of the magnetic stress vector is 
always T. The following construction (Fig. 5) is also useful : 
Let ABC be the plane on which the electric stress is required, 
BN the unit normal, BE the electric force, BP the stress. 



Then N, E, and P are in the same plane, and the angle be- 
tween N and E equals that between E and P. Or, the same 
operation which turns N to E also turns E to P, except as 
regards the tensor of P. 

To show the transition from a tension to aniequal pressure, 
imagine the plane ABC to be turned round, and with it the 
normal N. Of course E remains fixed, being the electric force 
at the point B. Start with coincidence of N and E. Then P 
also coincides with them, and represents a normal pull on the 
surface ABC. As N and E separate, so do E and P equally, 
so that when E makes an angle of 45 with N the normal pull 
is turned into a tangential pull, or a shearing stress, P being 


FIG. 5. 

now at right angles to N. Further increase in the angle E 
makes with N brings BP to the other side of BC j and when E 
is at right angles to N, we have P and N in the same line, but 
oppositely directed. That is, the tension has become converted 
into an equal pressure. 

Uncertainty regarding the General Application of the 
Electromagnetic Stress. 

73. We may now consider the practical meaning of the 
stress whose relation to the electric and magnetic forces has, 
unaer certain suppositions, been formularised. Go back to the 
foundation of electromagnetic theory, viz., the mechanical 
forces experienced by electrically charged bodies, by conductors 
supporting currents, and by magnets, intrinsic or induced. It 


is by observation of these forces in the first place, followed by 
the induction of the laws they obey, and then by deductive 
work, that the carving out Of space into tubes of force follows; 
and now, further, we see that the localisation of the stored 
energies, according to the square of the electric and magnetic 
force respectively, combined with the two circuital laws, leads 
definitely to a stress existing in the electromagnetic field, 
which is the natural concomitant of the stored energy, and 
which is the immediate cause of the mechanical forces observed 
in certain cases. But the theory of the stress goes so far be- 
yond experimental knowledge in some respects, although agree- 
ing with it in others, that we could only expect it to be true if 
the theoretical foundations were also rigidly true in all respects. 
Such is not the case, however. To begin with, the way of ex- 
pressing the action of ordinary matter merely by altering the 
values of the two ether constants, and by a fresh property, 
that of conductivity, is extremely bald. It is, indeed, surpris- 
ing what a variety of phenomena is explained by so crude a 

The objection is sometimes made against some modern theo- 
retical developments that they are complicated. Considered as 
an argument, the objection is valueless, and only worthy of 
superficial minds. Whatever do they expect ? Do they not 
know that experimental knowledge, even as at present existent, 
shows that the theory of electromagnetism, when matter is 
present, must, to be comprehensive, be something far more (in- 
stead of less) complicated than theory as now developed ? The 
latter is, as it were, merely a rough sketch of a most elabo- 
rate subject, only small parts of which can be seen at one 

In the next place, even if we take the stated influence of 
matter on the ether as sufficient for the purposes of a rough 
sketch, the theory of the stress should, except in certain rela- 
tively simple cases, be received with much caution. Why this 
should be so will be apparent on examining the manner in which 
the stress has been obtained from the circuital laws. If we inves- 
tigate the subject statically, and, starting from certain mechanical 
forces regarded as known, endeavour to arrive at a stress which 
shall explain those forces, we shall find that the problem is essen- 
tially an indeterminate one. All sorts of stress functions may 


be made up which are precisely equivalent in their effects in the 
gross, that is, as regards translating or rotating solid bodies 
placed in an electric or magnetic field. To remove this indeter- 
minateness a dynamical method must be adopted, wherein what 
goes on in the unit volume whilst its electric and magnetic 
states are changing, and the matter concerned is itself in motion, 
are considered. If our system of connections is dynamically 
complete and consistent, and is such that the flux of energy can 
be traced, then a determinate stress comes out, as we have 
found. The method is, at any rate, a correct one, however the 
results may require to be modified by alterations in the data. 
Besides that, the distribution of energy (electric and magnetic) in 
bodies is in some cases open to question ; and a really speculative 
datum is that concerning the motion of the ether as controlled 
by the motion of matter. Now, this datum appears to be one 
which is essential to the dynamical method ; the only alternative 
is the statical and quite indeterminate method. Our attitude 
towards the general application of the special form of the 
stress theory obtained should, therefore, be one of scientific 
scepticism. This should, however, be carefully distinguished 
from an obstinate prejudice founded upon ignorance, such as 
is displayed by some anti-Maxwellians, even towards parts of 
Maxwell's theory which have received experimental demon- 

The stress theory can, nevertheless, sometimes be received 
with considerable confidence, if not absolute certainty. The 
simplest case is that of ordinary electrostatics. 

The Electrostatic Stress in Air. 

74. Let there be no magnetic force at all, and the electric 
force be quite steady, and the medium be at rest, and there be 
no impressed forces. These limitations bring the circuital laws 
down to 

= *E, (1) 

-curlE = 0; ....'. (2) 

that is, there must be no conduction current anywhere, and 
the voltage in any circuit must be zero. The first condition 
(1) implies that there is no electric force in conductors. We 
may, therefore, divide space into conducting and non-conducting 


regions, and our electric field is entirely confined to the latter. 
The second condition (2) implies tangential continuity of E 
at the boundary of the non-conductor, so that as there is no E 
in the other or conducting side, there must be no tangential 
E on the non-conducting side. The lines of force, therefore, 
terminate perpendicularly on the conducting matter. Whether 
the conductors are also dielectrics or not is quite immaterial. 
The displacement also terminates normally on the conducting 
surface in the usual case of isotropy. Thus D, the tensor of D, 
measures the surface density of electrification, when the positive 
direction of D is from the conductor to the insulator. But in 
general it is the normal component of D, that is DN, where 
N is the unit normal vector, that measures the density of the 
electrification. Besides this, there may be interior electrification 
of the non-conductor, its volume density being measured by the 
divergence of the displacement. The arrangement of the 
electric force, so that the circuital voltage shall be zero 
throughout the non-conductor, and give the proper internal 
electrification, and the charges on the conductors, is uniquely 

We have thus the ordinary case of a number of charged con- 
ductors in air, with the difference that the air, or parts thereof, 
may be replaced by matter of different permittivity. It is also 
to be noted that one non-conducting region which is entirely 
separated from another by conducting matter may be taken by 
itself, and all the rest ignored. 

Now, first without replacing the air by matter of different 
permittivity, we see that there are two entirely different ways 
of considering the mutual actions of the conductors. The old 
way is analogous to Newton's way of expressing the fact of 
gravitation. We may say that any element of electrification p 
repels any other p' with a force 

pp'l 47rcr 2 , 

if r be the distance between the two charges, and that the 
resultant of all such forces makes up the real forcive. 

In the other way, appropriate to the philosophy of Faraday, 
as developed by Maxwell, these forces acting at a distance are 
mathematical abstractions only, and have no real existence. 
What is real is a stress in the electric field, of a peculiar nature, 


being a tension of amount U parallel to E, combined with an equal 
lateral pressure, and it is the action of this stress that causes 
the electrified conductors to move, or strains them, according to 
the way they are supported, when by constraints they cannot 
appreciably move. 

Since the electric force if normal to the conducting surfaces, 
the stress vector is entirely a normal pull of amount U per 
unit area, and the motions or tendencies to move of the con- 
ductors are perfectly accounted for by this pull. They do not 
move because of electrical forces acting at a distance across the 
air, but because they are subjected to moving force on the spot 
by the stress terminating upon them. 

Thus a charged soap-bubble is subjected to an external radial 
tension, and therefore expands; and so, no doubt, does a charged 
metal sphere to some small extent. The parallel plates of a 
condenser are pulled together. When they are very large 
compared with their distance apart, the force on either is 

J ED x area, 
= J E x charge. 

Here E is the transverse voltage divided by the distance be- 
tween the plates, so that, if the plates be connected to a constant 
.-source of voltage, the attraction varies inversely as the square 
-of the distance between them; whereas, if the plates be in- 
.sulated and their charges constant, the attraction is the same 
.at any distance sufficiently small compared with the size of the 

But by sufficiently separating the plates, or by using smaller 
^plates, the displacement, which was formerly almost entirely 
'between them, will spread out, and will terminate in appreci- 
able amount upon the sides remote from one another. By the 
pull on the remote sides thus produced the attraction will be 
lessened, and the further the plates are separated the more 
displacement goes to their backs, and the less is the attraction. 
When the distance is great enough it tends to be simply the 
-attraction between two point charges. Thus the attraction 
between two distant oppositely charged conducting spheres, 
which varies closely as the inverse square of the distance, 
depends entirely upon the slight departure from uniformity 
of distribution of the electrification over their surfaces, whereby 


the normal pull on either is made a little greater on the side 
next the other than on the remote side. Also, the inverse 
square law itself, which is exactly true for point charges, is 
merely the ultimate limit of this operation. 

Some attacks have been made on the law of inverse squares, 
especially in its magnetic aspect. But these attacks appear to 
have been founded upon misapprehension. The law is true, 
and always will be. 

The moving Force on Electrification, bodily and superficial. 

75. In the above electrostatic application of the stress, it 
will be observed that the tension alone comes into play, at 
least explicitly, owing to the tubes of displacement terminating 
perpendicularly on the conductors. Thus each tube may be 
compared with a rope in a state of tension, pulling whatever 
its ends may be attached to. But the lateral pressure is 
needed to keep the medium itself in equilibrium, so that the 
only places where translational force arises from the stress is 
where there is electrification. The mechanical force is 

F = EdivD = E/> ..... (3) 

per unit volume. This is the force on volume electrification,. 
and is the result of the differential action of the stress round 
about the electrification, as in the case of the inverse square 
law between point charges, lately mentioned. The correspond- 
ing surface force is 

F = iE.DN = NU ..... (4) 

per unit area. Now, here DN is the surface equivalent of 
div D, so there is at first sight a discrepancy between the ex- 
pressions for the force per unit volume (3), and per unit area 
(4), on bodily and surface electrification. How the coefficient 
J comes in may be seen by taking the limiting form of the 
previous expression. Let there be a thin skin of electrification, 
of amount o- per unit area ; E falling off from E outside to 
inside the skin. Evidently the mean E is JE, so that the 
total force on unit area of the skin obtained by summation 
of the forces on the volume electrification in the skin, is not. 
Eo-, but JEo-. This is merely a mathematical harmonisation^ 


From the point of view of the stress the difficulty does not pre- 
sent itself. 

The harmonisation is simply evident when the layer is of 
uniform density, for the electric force will then fall off in in- 
tensity uniformly. It might, however, be suspected that, per- 
haps, the result would not come out quite the same if we 
assumed any other law of distribution, and kept to it in pro- 
ceeding to the limit by making the skin infinitely thin. But a 
cursory examination will show that it is all right ; for if E is 
the electric force within the layer, the electrification density 
will be c (dE/dx) and the translational force will be cE (dE/dx) 
per unit volume, if x is measured perpendicularly to the skin's 
surfaces. Integrate through the skin, and the result is 

where the suffixes refer to the value just outside the skin, on 
its two sides. In the present case the second term U 2 is zero 
(within the conductor), so that the result is the single normal 
pull of the tension on the non-conducting side. 

Depth of Electrified Layer on a Conductor. 

76. In this connection the old question of the depth of the 
layer of electrification on a conducting surface crops up. Has 
it any depth at all, and, if so, how much ? The question is not 
so superficial as it looks, arid the answer thereto lies in the 
application thereof. If a powerful mental microscope be applied 
to magnify the molecules and produce evident heterogeneity, 
the surface of a conductor would become indefinite ; and unless 
the molecules were found to be very closely packed, it is evident 
that the displacement in the ether outside the conductor would 
not terminate entirely upon those molecules which happen to 
be most superficially situated, but that a portion of the dis- 
placement would go deeper and in sensible amount reach 
molecules beneath the first set, and an insensible amount might 
penetrate through many layers. Thus in a molecular theory 
the depth of the layer of electrification has meaning, and could 
be roughly estimated. 

But the case is entirely different in a theory which delibe- 
rately ignores molecules, and assumes continuity of structure. 


A conductor is then a conductor all through, and not a 
heterogeneous mixture ; and the surface of a conductor is an 
unbroken surface. The electrification on it is therefore surface 
electrification, and has no depth. For it to be otherwise is 
simply to make nonsense. It is desirable to be consistent in 
working out a theory, for the sake of distinctness of ideas ; if, 
then, we wish to give depth to surface electrification, and still 
keep in harmony with Maxwell's theory, we must change our 
way of regarding a conductor, and bring in heterogeneity. 
Each view is true, in its own way ; but as in the mathematical 
theory continuity of structure is tacitly assumed, we have a 
simultaneous evanescence of one dimension in the distribution 
of electrification. 

The same question occurs in another form in the estimation 
of -bodily electrification, when the meaning of volume density 
of electrification is considered. When air is electrified, it is 
probable that the electrification is carried upon the foreign 
particles suspended in the air, and it may be partly upon the 
air molecules themselves. In either case it is ultimately surface 
electrification, and quite discontinuous. But, merely for the 
sake of facility of working, it is desirable to ignore all the dis- 
continuity, and assume a continuous and practically equivalent 
distribution of bodily electrification. Thus, as previously we 
saw surface density to be a kind of volume density, so now we 
see that volume density is a kind of surface density. When, 
therefore, we say that the translational force per unit volume 
is E/>, where E is the electric force and p the volume density of 
electrification, we really mean that E/> is the average, obtained 
by summation, of the translational forces on the multitudinous 
electrified particles, every one of these forces being itself a 
differential effect, as before seen, viz., the resultant of the 
unequal pulls on different parts of a particle exerted by the 
electric stress. 

As ether has some of the properties of matter, and as electri- 
fication is found in association with matter, it is possible, 
however improbable, that ether itself may become electrified. 
But of this nothing is known. Nor, more importantly, is it 
understood why the electric stress appears to act differently on 
positively and on negatively electrified matter. But, if we 
begin to talk about what is not understood, we enter illimit- 


able regions. Men who are engaged in expounding practical 
problems sometimes make the boast that they take and discuss 
things as they are, not as they might be. There is a sound of 
specious plausibility here, which is grateful and comforting ; 
but, as a matter of plain fact, questions of physics never are 
theorised upon as they are, but always as they might be. It is 
a necessity to limit the field of inquiry, for to take things as 
they are, or as they seem to be, would lead at once to a com- 
plete tangle. For the problem, as it presents itself in reality, 
there is always substituted a far simpler one, containing certain 
features of the real one emphasised, as it were, and othera 
altogether omitted. The juveniles, who take things as they 
are, do not do it ; they only think so. They may strain out a 
few gnats successfully, but swallow, quite unawares, all the 
camels in Arabia. But the principle and practice of limitation 
and substitution is the same all over; in politics, for instance, 
where a fictitious British Constitution does brave duty, as a 
scarecrow, and in other useful ways. 

Electric Field disturbed by Foreign Body. Effect of a. 
Spherical Non-conductor. 

77. To further exemplify the significance of the electric 
stress, let us introduce a foreign body into a stationary electric 
field. The field will be disturbed by its introduction, and will 
settle down to a new state ; the change depending upon the 
nature of the foreign body, whether conducting or non-conduct- 
ing, in substance or superficially, and upon whether it has a 
charge itself, or contains any other source of displacement. If 
it be a good conductor, either charged or uncharged, the final, 
state, reached very quickly, will be such that the displacement 
will terminate normally upon its surface, thus reproducing the 
previous case ( 74 to 76). But if it be a non-conductor, the 
result is somewhat different. If superficially conducting, we 
may indeed have an ultimate electrification of the surface, so as 
to come wholly or partly under the same case : but if there be 
no superficial or internal conduction, or only so little that a 
long time must elapse for it to become fully effective, what we 
do is simply to replace the dielectric air in a certain region by 
another dielectric of different permittivity, usually greater.. 


Then, supposing the external field to be due to charges upon 
insulated conductors or to internal electrification in the air 
(kept at rest for the purposes of the argument), no change will 
occur in their amounts ; but there will be merely an alteration 
in the distribution of the electrification on the conductors, 
caused by the displacement becoming denser within the foreign 
body than before (or less dense, if its permittivity can be less 
than that of the air). The intrinsic electrification of the body 
itself, if any, must also be allowed for; but should it have 
none previously, it will remain unelectrified when introduced 
into the field, and the displacement will pass freely through it, 
and out again in a solenoidal manner. This is expressed by 
the surface condition 

where D : and D 2 are the displacements in the air and foreign 
body respectively at their interface, and N 1? N 2 are the unit 
normals from the interface to the two media. 

The only quite simple case (excepting that of infinite plane 
sheets) is that of a sphere of uniform permittivity brought into 
a previously uniform field. The ultimate displacement in the 
sphere is then parallel to the original displacement in the air. 
It may vary between zero and three times the original displace- 
ment, as the permittivity of the sphere varies from zero up to 
infinity. It is certainly a little surprising that the ultimate 
displacement with infinite permittivity should be only three 
times the original (and it is not much less when the permittivity 
is only 10 times that of the air) ; whilst, on the other hand, the 
zero displacement when the permittivity is zero (a quite ideal 
case) is obvious enough, because the displacement never enters 
the sphere at all, but goes round it. The original uniform 
field may be conveniently that between the parallel plates of a 
very large air condenser. There is, however, a double action 
taking place. When the transverse voltage of the condenser is 
maintained constant by connection with a suitable source, the 
insertion of the foreign body, which increases or reduces the 
permittance of the condenser, will increase or reduce its charge 
under the action of the constant source, besides concentrating 
the displacement within the body, or the reverse. With constant 
charges, however, when the plates are insulated, the insertion 


of the foreign body will reduce the transverse voltage when its 
permittivity exceeds that of the air ; and conversely. Increase 
of permittivity also increases the stored energy when the trans- 
verse voltage is constant, but reduces it when the charges are 

Now, since the electric force does not terminate on the boun- 
dary of the foreign body, but extends all through it, so does the 
electric stress. So far, however, as the resultant force and 
torque on the body, when solid, is concerned, we may ignore 
the internal stress altogether, and consider only the external, or 
stress in the air. This is a particular case of a somewhat impor- 
tant and wide prope^ in abstract dynamics, which we may 
state separately thus. 

Dynamical Principle. Any Stress Self-equilibrating. 

78. The resultant force and torque due to any stress in any 
region is zero. Or, any stress in any region forms a self-equili- 
brating system. 

Imagine any distribution of stress to exist in a region A, and 
to terminate abruptly on its boundary. Or, equivalently, 
imagine a piece of a stressed solid to be removed from its place 
without altering the stress. The stress-variation, when esti- 
mated in a certain way, constitutes mechanical force tending to 
move the body. This will be, in the case of an ordinary irrota- 
tional stress, entirely translational force. But there are two kinds. 
First, there is internal force, reckoned per unit volume, due to 
:he continuous variation of the stress in the body. Next, 
;here is superficial force, reckoned per unit area, due to the 
abrupt cessation of the stress. This surface traction is repre- 
sented simply by the stress vector itself, acting on the inner 
side of the surface of the body. Now, the resultant effect of 
these two forcives, over the surface and throughout the volume 
of the solid respectively, in tending to translate and rotate it, 
is zero. Or, in other words, the force and torque equivalent to the 
surface forcive are the negatives of those due to the internal 
forcive. If it were otherwise, the differential action would 
cause indefinite increase in the translational and rotational 
energy to arise out of the internal mutual forces only of a 


If the stress be of the rotational type, there will be an in- 
ternal torque (per unit volume) as well as a translational 
force. Still, however, the resultant force and torque due to 
the surface tractions will cancel those due to the internal forces 
and torques. 

In case there be any difficulty in conceiving the traction exerted 
on the surface of a body by the stress within itself, we may 
replace the sudden cessation of the stress by a gradual cessation 
through a thin skin. The solid is then under the influence of 
continuous bodily force only (and torque also, if the stress be 
rotational) conveniently divisible into the internal force all 
over, and the force in the skin. Otherwise it is the same. 

Now put the solid piece back into its place again. Since its 
own stress balances itself, we see that whatever the forcive on 
the piece may be it must be statically equivalent to the action 
upon its boundary of the external stress only, constituting an 
external surface traction. There need be no connection between 
the external and the internal stress. The latter may be any- 
thing we like, so far as the resultant force and torque on the 
piece are concerned. The difference will arise in the strains 
produced, or in the relative internal motions, when for one stress 
another is substituted*- 

Electric Application of the Principle. Resultant Action on 
Solid Body independent of the Internal Stress, which 
is statically indeterminate. Real Surface Traction is 
the Stress Difference. 

79. Returning to the electric field, we see that whatever 
be the nature of the reaction of the foreign body on the original 
state of the field, the resultant mechanical action on the body 
as a whole is fully represented by the stress in the air just 
outside it, in its actual state, as modified by the presence of 
the body, and that we need not concern ourselves with the 
internal state of stress. Nor are we limited, in this respect, by 
any assumed proportionality of electric force to displacement 
in the body, or assumed absence of absorption, or other irregu- 
larities and complications. That is, we need not have any 
theory to explicitly account for the change made in the electric 
field by the body. Nor do we gain any information regarding 


the internal stress from merely a knowledge of the external 
stress, although that involves the reaction of the body on the 
electric field. This is the meaning of the statical indeter- 
minateness of the stress before referred to, and the principle 
applies generally. 

The air stress vector P will usually have both a normal and 
a tangential component at the surface of a body, viz.: 

U cos 26 normal, 
U sin 28 tangential, 

if be the angle between the normal N to the surface (drawn 
from the body to the air) and the electric force E, and U the 
density of the electric energy, or the tensor of the stress vector, 
In only one case, however, will this external surface traction 
represent the real forcive in detail (as well as in the lump), 
viz., when there is no stress at all on the other side of the 
boundary, that is, in the case of a conductor in static equilibrium. 
In general, the real surface force is represented by the vector 
P 1 -P 2 , the stress difference at the boundary, P : being the 
external and P 2 the internal stress vector, which two stresses 
may, if we please, be imagined to be united continuously by a 
gradually changing intermediate stress existing in a thin skin, 
an idea appropriate to molecular theories. (It may be remem- 
bered that P when positive means a pull.) Each of these may 
be split into a normal and a tangential component. Now, the 
tangential components are 

Uj sin 20j and U 2 sin 20 2 , 

where the suffix x relates to the air, and 2 to the other medium. 
Or E x Dj sin l cos 6 l and E 2 D 2 sin 2 cos 6%, 

But here we have normal continuity of the flux D, and tangential 
continuity of the force E; (otherwise the surface would be 
electrified and covered with a magnetic current sheet); that is 

D 1 cos^ 1 = D 2 cos #2, ) 
and E 1 sin^=E 2 sm0 2 . / 

These relations make the tangential tractions equal and 
opposite, so that there is no resultant tangential traction, and 



the actual traction is entirely normal, being the difference of the 
normal components of P l and P 2 , or, 

.... (6) 
which is the same as (being subject to (5),) 

^D x cos O l (E x cos 1 - E 2 cos 2 ) 
- P! sin ^ (D x sin 6^-0,3 sin 2 ) . . . (7) 
The coefficient J comes in for a similar reason to before, 75. 

This formula (6) or (7) being the real surface traction when 
E varies as D in the body, and the stress is of the same type as 
in air, furnishes a second way of calculating the resultant force 
and torque on the body, when its permittivity is uniform ; and 
it is noteworthy that the surface traction is, as in the case of an 
electrified conductor in equilibrium, entirely normal, although 
it may now be either a pull or a push. 

Translational Force due to Variation of Permittivity. 
Harmonisation with Surface Traction. 

80. Noting that the mechanical force on the elastically 
electrizable body is situated where the change of permittivity 
occurs, and is in the direction of this change, it may be inferred 
that when the permittivity varies continuously there is a bodily 
translational force due to the stress variation which is in the 
direction of the most rapid change of permittivity. This is, in 
fact, what the stress vector indicates when c varies continuously, 
viz., the force represented by 

-JEV=-VJJ, ..... (8) 

where yc means the vector rate of fastest increase of c round- 
about the point considered. Since U = J C E 2 , the second form 
in (8) will be understood, meaning the vector- slope of U as 
dependent upon the variation of c only. 

This bodily force, and the previous surface force may be har- 
monised by letting c vary not abruptly, but continuously from 
the value Cj on one side to c 2 on the other side of the surface of 
discontinuity, through a thin skin, and summing up the trans- 
lational forces in the skin by the formula (8). Thus, if x be 


measured normal to the skin, the translational force per unit 
volume is - JE 2 (dc/dx) in the direction of the normal ; or 

, dc 

~ 2-kf -j- 

dx dx 

where E ft and D n are the normal components of E and D, and 
E 4 the tangential component of E. Now D n and E f are con- 
stant ultimately, for the reason before given, so we can inte- 
grate (9) with respect to x immediately, giving 

..r id*.- 

between the limits ; or, since E n and E t are proportional to the 
cosine and the sine of 6 respectively. 

[JcE 2 cos2<9] ..... (11) 

between the limits, which is the same as (6), which was to be 

Movement of Insulators in Electric Field. Effect on the 
Stored Energy. 

81. Since a sphere of uniform permittivity placed in a uniform 
field causes the external lines of electric force to be symmetrically 
distorted fore and aft, it has no tendency to move, but is merely 
strained. But if the body be not in an initially uniform field, 
or be not spherical, complex calculations are usually needed 
to determine the effect. If, however, it be only a small piece, 
the tendency is for it to move in the direction in which the 
energy, or the stress, in the field increases most rapidly, inde- 
pendent of the direction of the electric force, when its permit- 
tivity exceeds that of the gaseous medium. The total electric 
energy will be diminished by permitting the motion when the 
charges are constant ; but increased should the field be kept up 
by constant sources of voltage. 

These properties are rendered particularly evident by taking 
the extreme cases of infinite and zero permittivity of a small 
body placed in a widely varying field, that surrounding a charged 
sphere, for example, the electric force varying in intensity as 

H 2 


the inverse square of the distance from its centre. Here the 
non-permittive body is repelled, for the lines of force go round 
it, and the lateral pressure comes fully into play, and is greater 
on the side next the charged sphere. On the other hand, with 
the infinitely permittive body, concentrating the displacement, 
it is the tension that comes fully into play, and this being 
greater on the side next the charged sphere, the result is an 
attraction. The permittance of the sphere, also, is increased in 
the latter case, and decreased in the former that is, when the 
natural motion of the body to or from the sphere is allowed ; 
so, since the total electric energy is JSV 2 where S is the permit- 
tance and V the voltage, or, equivalently, JVQ, if Q is the 
charge, we have always a diminution of energy when the 
natural motion is allowed, whether resulting from attraction or 
repulsion, if the charge is constant ; but an increase of energy 
if the voltage is constant. 

In intermediate cases the tension is dominant when the c of 
the body exceeds, and the pressure is dominant when it is less 
than that of the air, there being perfect equilibrium of a piece 
of any shape in any field if there be equality of permittivity, 
and therefore no disturbance of the field. Here we see the part 
played by the lateral pressure in the case of conductors in 
equilibrium. It has no influence on them immediately, and 
might be thought wholly unnecessary, but it is equally 
important with the tension in the non-conducting dielectric 

Magnetic Stress. Force due to Abrupt or Gradual Change of 
Inductivity. Movement of Elastically Magnetised Bodies. 

82. Passing now to the corresponding magnetic side of the 
stress question, we may observe that the analogy is an imperfect 
one. Thus, proceeding as at the beginning of 74, to have a 
stationary magnetic field without impressed forces, we shall 
find that there must first be no magnetic conduction current ; 
and next, that the gaussage in any circuit must be zero. The 
magnetic force we then conclude to be confined entirely to the 
magnetically non-conducting regions, and to terminate perpen- 
dicularly upon their boundaries. Thus we come to the 
conception of a number of detached magnetic conductors im- 


mersed in a magnetically non-conductive medium, these con- 
ductors having magnetic charges on them measured by the 
amount of induction leaving or terminating upon them, with 
possible associated volume magnetification in the non-conduct- 
ing medium. 

The magnetic stress will exert a normal traction 

. . , . . (12) 
per unit area on the conductors, and a force 

Hcr = Hdiv.B ..... (13) 

per unit volume on tridimensional magnetification, of density cr, 
measured by the divergence of the induction. These are analo- 
gous to the forces on surface and volume electrification. 

Also, when the inductivity /* varies, we shall have a normal 
surface traction of amount, 

T 1 cos2^ 1 -T 2 cos2^, .... (14) 

per unit area, analogous to (6), when p changes value abruptly 
at the interface of two media ; and a force 

-iH 2 W ...... (15) 

per unit volume, when /z varies continuously. 

The forces (12), (13), however, are absent, because of the ab- 
sence of magnetic conductivity, and, in connection therewith, 
the absence of " magnetification." But (12) may be sometimes 
used, nevertheless, when it is the stress across any surface that 
is in question, and we create surface magnetification by regard- 
ing one side only. 

We are, therefore, left with the forces (14), (15), depending 
upon variation of inductivity, abrupt or gradual. These ex- 
plain the mechanical action upon elastically magnetised 
media, e.g., the motions of bodies to or from a magnet pole, 
according as they are paramagnetic or diamagnetic, which, it 
should be remembered, depends fundamentally upon the varia- 
tion in the intensity of magnetic force near the pole ; and the 
axial equilibrium of a paramagnetic bar, and equatorial equi- 
librium of a diamagnetic bar. Faraday's remarkable sagacity 
led him to the essence of the explanation of these and other 
allied phenomena, as was later mathematically demonstrated by 


Sir W. Thomson. Questions relating to diamagnetic polarity 
are, in comparison, mere trifling. 

Force on Electric Current Conductors. The Lateral Pressure 
becomes prominent, but no Stress Discontinuity in general. 

82a. But the magnetic stress has other work to do than to- 
move elastically magnetised matter under the circumstances 
stated. Wholly independent of magnetisation, it produces the 
very important moving force on conductors supporting electric- 
current, first mathematically investigated by Ampere. 

The lateral pressure of the stress here comes prominently 
into view, when we ignore the stress in the interior of the con- 
ductors, so that the stress vector in the air at the boundary of 
a conductor represents the moving force on it per unit area. 
Thus, two parallel conducting wires supporting similar currents 
attract one another, because their magnetic forces are additive 
on the sides remote from one another, rendering the lateral 
pressure on them greater there than on the sides in proximity. 
But when the currents are dissimilar the magnetic force is- 
greater on the sides in proximity, and, therefore, the lateral 
pressure of the stress is greater there, producing repulsion. 

Proceeding further, and considering the stress within ih& 
conductor also, according to the same law, we find this pecu- 
liarity. In the case of unmagnetisable conductors (typified 
practically by copper), there is no superficial discontinuity in 
the stress, and therefore no surface forcive of the kind stated. 
This may be easily seen from 79, translating the results from 
the electric to the magnetic stress. There is no tangential 
discontinuity in the stress because the normal induction and 
tangential magnetic force are continuous ; and there is no dis- 
continuity in the normal component of the stress because (since 
there is no difference of inductivity) the normal magnetic force 
and the tangential induction are also continuous. 

In (5), (6), (7) turn E to H, and D to B, and U to T, and 
note that (5), as transformed, are true when the force and flux 
are exchanged, so that the transformed expressions (6) or (7) 

In the case of a real conductor, therefore, with finite volume 
density of electric current, the moving force is distributed 


throughout its substance ; and the variation of the stress indi- 
cates that the translational force per unit volume is expressed 

by F = VOB, . ..... . (16) 

where is the current density. This is what Maxwell termed 
"the electromagnetic force," and it is what is so extensively 
made use of by engineers in their dynamos, motors, and things 
of that sort. It is probable, I think, that in grinding away at 
the ether they also stir it about a good deal, though not fast 
enough to produce sensible disturbances due to etherial dis- 

Force on Intrinsically Magnetised Matter. Difficulty. Max- 
well's Solution probably wrong. Special Estimation of 
Energy of a Magnet and the Moving Force it leads to. 

83. There is next the force on intrinsically magnetised 
matter to be considered in connection with the magnetic stress. 
This is, perhaps, the most difficult part of magnetic science. 
Although, so far as the resultant effect on a magnet is con- 
cerned, we need not trouble about its internal state, but, as 
before, merely regard it as being pushed or pulled by the stress 
in the surrounding air, such stress being calculable from the 
distribution of magnetic force immediately outside it, as 
modified by the magnet itself, yet it is impossible that the real 
forcive can be represented merely by the surface traction P N . 

Now, it is possible to find a distribution of magnetification 
over the surface, which shall be externally equivalent to the 
interior magnetisation, or to whatever other source of induction 
there may be. Then we may substitute for the surface trac- 
tion P N , another traction, namely, upon the surface magnetifi- 

Or, we may find a distribution of fictitious electric current 
upon the boundary of the magnet, which shall be externally 
equivalent to the interior sources, and then represent the 
forcive by means of fictitious electromagnetic force on this 
current, 82a. 

Or we may combine these methods in various ways. Evi- 
dently, however, such methods are purely artificial, and that 
to obtain the real forcive we must go inside the magnet. This 
can only be done hypothetically, and with precarious validity. 


One way of exhibiting the mechanical action on a magnet, or 
on magnetised matter generally, is that given by Maxwell in his 
chapter on the stresses (Vol. II.). This I believe to be quite 
erroneous for many reasons, the principal being that it does 
not harmonise with his scheme generally, and that it lumps to- 
gether intrinsic and induced magnetisation, which have essen- 
tial differences and are physically distinct. There are many 
other ways of exhibiting the resultant force and torque as made 
up of elementary forces, acting upon magnetisation, or on free 
magnetism, or on the variation of magnetism estimated in 
different ways. It is unnecessary to enter into detail regarding 
them. Nobody would read it. It will be sufficient to point out 
the particular way which harmonises with Maxwell's scheme in 
general, in the form in which I display it, with a special esti- 
mation of magnetic energy. Proportionality of force and flux 
is assumed. The want of this proportionality is quite a separate 
question. Given a definite relation between force and flux, the 
accompanying change in the stress vector, in accordance with 
the continuity of energy, can be estimated. This I have re- 
cently shown how to do in another place, which shall not be 
more explicitly referred to. 

Intrinsic magnetisation possesses the peculiarity that it is, in 
a manner, outside the dynamical system formulated in the 
electromagnetic equations, inasmuch as it needs to be ex- 
hibited in them through the medium of an impressed force, 
although this is disguised in the ordinary mode of representa- 
tion. Calling this intrinsic magnetic force h (any distribution), 
as before, the induction due to it in a medium of any inductivity 
(varying continuously or abruptly, if required) is found in the 
same way as the displacement due to intrinsic electric force in 
a non-conducting medium of similarly distributed permittivity ; 
or as the conduction current due to the same in a medium of 
similar conductivity ; or, to make a fourfold analogy, as the 
magnetic conduction current due to h in a (fictitious) medium 
of similar magnetic conductivity. 

I may here point out that a clear recognition of the correct 
analogies between the electric and magnetic sides of electro- 
magnetism is essential to permanently useful work. Many 
have been misled in this respect, especially in comparing 
Maxwell's displacement with magnetic polarisation. The true 


analogue of D is B. Investigations based upon the false 
foundation mentioned can lead to nothing but confusion. 
There are enough sources of error without bringing in gratuit- 
ous ones. 

Now, presuming we have induction set up by h , how is the 
energy to be reckoned ? I reckon its amount per unit volume 
to be JHB generally, whether outside or within the magnet ; 
or, in the usual isotropic case, J/*.H 2 or J/ir^B 2 . This makes 
the total work done by h to be h B per unit volume, if h is 
suddenly established ; of which one half is wasted, and the 
rest remains as stored magnetic energy. That is, 

, .... (17) 

if the 2 indicates space-summation. If h be suddenly des- 
troyed, the energy 2T is set free and is dissipated, mainly by 
the heat of currents induced within the magnet itself and sur- 
rounding conductors. This is the meaning of JHB being the 
stored energy per unit volume. But it may not be immediately 
available. To take an extreme case, if we have a complete 
magnetic circuit, so magnetised intrinsically that there is no 
external field, the energy, as above reckoned, is the greatest 
possible, since H = h . But it is now not at all available, unless 
h be destroyed. On the other hand, Maxwell would appear to 
have considered the energy of a magnet to be 2 J/*(H - h ) 2 , 
which is zero in the just mentioned case. This reckoning does 
not harmonise with the continuity of energy, although it has 
significance, considered as energy more or less immediately 
available without destruction of h . The connection of the two 
reckonings is shown by 

lio); . (18) j 
or, when H and B are parallel, 

2|h B = 2i J uh 2-2^(H-li )2. . . (19). 

Now, according to the reckoning (17) of the stored energy, 
and the consequent flux of energy, the stress vector derived 
therefrom indicates that the moving force per unit volume is 

F = Vj B ....... (20), 

where j = curl h n . . . , , . . (21). 


The interpretation is that the vector J represents the distri- 
bution of (fictitious) electric current, which would, under the 
same circumstances as regards inductivity, set up, or be asso- 
ciated with, the same distribution of induction B as the intrinsic 
magnet is. It may be remarked that the induction due to an 
intrinsic magnetic force does not depend upon its distribution 
primarily, but solely upon that of its curl. 

Substituting this current system for the intrinsic magnetise 
tion, equation (20) indicates that the moving force is " the 
electromagnetic force " corresponding thereto, according to (16), 
82. The accompanying surface distribution of current, repre- 
senting the abrupt cessation of h , must not be forgotten; or 
we may let it cease gradually, and have a current layer in the 
skin of the magnet. It may be far more important than 
the internal current, which may, indeed, be non-existent. 
For instance, in the case of a uniformly longitudinally mag- 
netised bar, the equivalent current forms a cylindrical sheet 
round the magnet. In general, the surface representative of 

Jo is 

VNh , (22) 

where N is a unit normal drawn from the boundary into the 
magnet. It should also not be forgotten that if the inductivity 
changes, there is also the moving force (15) or (14) to be 
reckoned, besides that dependent upon the intrinsic magnetisa- 

Force on Intrinsically Electrized Matter. 

84. The electric analogue of intrinsic magnetisation is in- 
trinsic electrisation, represented in a solid dielectric in which 
"absorption " has occurred, and perhaps in pyroelectric crystals. 
It seems very probable that there is a true electrisation, quite 
apart from complications due to conduction, surface actions, 
and electrolysis. When formulated in a similar manner to in- 
trinsic magnetisation, by means of intrinsic electric force e 0> 
producing displacement according to the permittivity of the 
medium (which displacement, however, is now also affected by 
the presence of conducting matter), and with a similar reckon- 
ing of the stored energy, viz., JcE 2 per unit volume, where R 


includes e , we may expect to find, and do find, a moving force 
analogous to (20), viz., 


where g = - curl e . ... . . . (24). 

That is, magnetoelectric force on the fictitious magnetic cur- 
rent g which is equivalent to the intrinsic force e . And, similarly 
to before, we may remark that the flux due to e depends solely 
upon its curl. 

Summary of the Forces. Extension to include varying States 
in a Moving Medium. 

85. Now bring together the different moving forces we have 
gone over. On the electric side we have 

, . (25), 
and on the magnetic side 

F m = [H.r]-iH2v M + VOB + Vj B, . . (26), 

where the third term on the right of (25), and the first on the 
right of (26), in square brackets, are zero ; Ho- being force on 
magnetification and VDK the magnetic analogue of VOB. 
The other component forces can all separately exist, and in 
stationary states. 

Passing to unrestricted variable states, with motion of the 
flux-supporting media also, the electric and magnetic stress 
vectors indicate that the moving forces arising therefrom are 
obtainable from those exhibited in (25), (26), by simply 
changing the meaning of the electric current and the 
magnetic current symbols in the third terms on the right. 
Above, they stand for conduction current only, and one of 
them is fictitious. They must be altered to G and J , the 
" true " currents, as explained in 66. Thus 

. . (27) 
. . (28) 

express the complete translational forces due to the electric and 
magnetic stresses (31), (32), 72. 


The division of the resultant translational force into a 
number of distinct forces is sometimes useless and artificial. 
But as many of them can be isolated, and studied separately, 
it is not desirable to overlook the division. 

The equilibrium of a dielectric medium free from electrifica- 
tion and intrinsic forces, which obtains when the electric and 
magnetic forces are steady, is upset when they vary. There is 
then the electromagnetic force in virtue of the displacement 
(electric) current and the magnetoelectric force in virtue of the 
magnetic current ; that is, 

F = VDB + VDB (29) 

dt v* dt 

where W is the flux of energy. Here we neglect possible small 
terms depending on the motion of the medium. 

That there should be, in a material dielectric, moving force 
brought into play under the action of varying displacement and 
induction does not present any improbability. But it is less 
easy to grasp the idea when it is the ether itself that is the 
dielectric concerned. Perhaps this is, for some people, because 
of old associations the elastic solid theory of light, for in- 
stance, wherein displacement of the ether represents the 

But if we take an all round view of the electromagnetic con- 
nections and their consequences, the idea of moving force on 
the ether when its electromagnetic state is changing will be 
found to be quite natural, if not imperatively necessary. We 
do know something about how disturbances are propagated 
through the ether, and we can, on the same principles, allow 
for bodily motion of the ether itself. Further, reactions on the 
ether, tending to move it, are indicated. But -here we are 
stopped. We have no knowledge of the density of the ether, 
nor of its mechanical properties in bulk, so, from default of real 
data, are unable to say, except upon speculative data, what 
motions actually result, and whether they make any sensible 
difference in phenomena calculated on the supposition that the 
ether is fixed. 


Union of Electric and Magnetic to produce Electromagnetic 
Stress. Principal Axes. 

86. From the last formula we see that to have moving force 
in a non-conductor (free from electrification, &c.) requires not 
merely the coexistence of electric and magnetic force, but also 
that one or other of them, or both, should be varying with the 
time. That is, when the energy flux is steady, there is no 
moving force, but when it varies, its vector time-variation, 
divided by -y 2 , expresses the moving force. 

The direction of W is a natural one to choose as one of the 
axes of reference of the stress, being perpendicular to both E 
and H, which indicate the axes of symmetry of the electric and 
magnetic stresses. The two lateral pressures combine together 
to produce a stress on the W plane 

Pw^-W^U + T), .... (31) 

where W x is a unit vector parallel to W. (Take N = W l in the 
general formula (30) 72 for P N ). 

If, further, E and II are perpendicular to one another, the 
stresses on the planes perpendicular to them are 

P E1 =E 1 (U-T) J (32) 

P^H^T-U), (33) 

which are also entirely normal. Here E L and H x are unit 
vectors. Thus, W x is always a principal stress axis, while 
E : and Hj_ become the other pair of principal axes when they 
are perpendicular, as in various cases of electromagnetic waves. 
Thus when a long straight wire supports a steady current, or 
else is transmitting waves, the principal axes of the stress at a 
point near the wire are respectively parallel to it and perpen- 
dicular to it, radially and circularly. The first one has a 
pressure (U + T) acting along it, the second (parallel to E) a 
pressure (T U), and the third (parallel to H) a pressure 
(U-T). (This legitimate use of pressure must not be con- 
founded with the utterly vicious misuse of pressure to indi- 
cate E.M.F. or voltage, by men who are old enough to know 
oetter, and do.) In general, U and T are unequal. But in the 


case of a solitary wave or train of waves with negligible distor- 
tion U and T are equal, and there is but one principal stress, 
viz., that with axis parallel to W, a pressure 2U or 2T. It is 
this pressure (or its mean value) that is referred to as the 
pressure exerted by solar radiation, and its space-variation con- 
.stitutes the moving force before mentioned. This matter is 
.still in a somewhat speculative stage. 

It is natural to ask what part do the stresses play in the 
propagation of disturbances ? The stresses and accompanying 
strains in an elastic body are materially concerned in the trans- 
mission of motion through them, and it might be thought that 
it would be the same here. But it does not appear to be so 
from the electromagnetic equations and their dynamical con- 
. sequences that is to say, we represent the propagation of dis- 
turbances by particular relations between the space- and the 
time-variations of and H ; and the electromagnetic stress 
and possible bodily motions seem to be accompaniments rather 
than the main theme. 

Dependence of the Fluxes due to an Impressed Forcive upon 
its Curl only. General Demonstration of this Property. 

87. In 83 it was remarked that the flux induction due to 
an intrinsic magnetic forcive depends not upon itself directly, 
but upon its curl, and in 84 a similar property was pointed 
out connecting the flux displacement and intrinsic electric 
force. That is to say, the fluxes depend upon the vectors J 
and g , not upon e and h . This remarkable and, at first 
sight, strange property, which is general, admits of being 
demonstrated in a manner which shall make its truth evident 
in a wider sense, and lead to a connected property of consider- 
able importance in the theory of electromagnetic waves. 

Let there be any impressed forcive e in a stationary medium. 
Its activity is e J per unit volume, where J is the electric cur- 
rent, and the total activity is 2e J, where the 2 indicates sum- 
mation through all space, or at any rate so far as to include 
every place where e exists. Its equivalent is Q , the total rate 
of waste of energy, and the rate of increase of the total stored 
energies, say U and T . Thus, 

T (34). 


Now, the value of the summation is zero if e has no curl, or 
is irrotational that is, an irrotational forcive does no work 
upon a circuital flux. This proposition may be rendered evident 
by employing a particular method of effecting the space sum- 
mation, viz., instead of the Cartesian method of cubic sub- 
division of space, or any method employing co-ordinates, divide 
space into the elementary circuital tubes belonging to the flux. 
(Here it is J.) Fixing the attention upon any one of these 
circuits, in which the flux is a constant quantity, we see at 
once that the part of the summation belonging to it is the 
product of the impressed voltage in the circuit and the flux 
therein. But there is no impressed voltage, because e has no 
curl ; hence the circuit contributes nothing to the summation. 
Further, since this is true for every one of the elementary cir- 
cuits, and inclusion of them all includes all space, we see that 
the summation 2e J necessarily vanishes under the circum- 
stance stated of e being irrotational. 

Now return to equation (34), and suppose that the initial 
state of things is absence of E and H everywhere, so that Q , U 
and T are all zero. Next start any irrotational e , and see 
what will happen. The left side of (34) being zero, the right 
side must also be and continue zero. But Q , U and T when 
not zero are essentially positive. Now if (U + T ) becomes 
positive, Qo should become negative, in order to keep the right 
number of (34) at zero. This negativity of Q being impos- 
sible, these quantities Q , U and T must all remain zero. Con- 
sequently E and H must remain zero. That is, an irrotational 
forcive can produce no fluxes at all, if the flux corresponding 
to the force is restricted to be circuital. 

It should be observed that this proposition applies not merely 
to the steady distribution appropriate to the given forcive, but 
to all intermediate stages, involving both electric and magnetic 
force, and flux of energy. Nothing happens, in fact, when any 
distribution of impressed force is made to vary in time and in 
space, provided it be restricted to be of the irrotational type, 
so that the voltage (or the gaussage) in every circuit is nil. 

Notice further the dependence of the property upon cir- 
cuitality of the factor with which the impressed force is 
associated (thus J with e , and similarly G with h ), and the 
positivity of energy ; and, more strikingly, the independence 



CH. II. 

of such details as are not concerned in equation (34), such as 
the distribution of conductivity, permittivity, &c., or of the 
forces being linear functions of the fluxes. 

Identity of the Disturbances due to Impressed Forcives 
having the same Curl. Example: A Single Circuital 
Source of Disturbance. 

. 88. Returning to (34) again, we see that any two impressed 
forcives produce the same results in every particular, as regards 
the varying states of and H gone through, if their curl is 
the same. For the difference of the two forcives is an irrota- 
tional forcive, and is inoperative. Here it is desirable to take 

FIG. 6. 

an explicit example for illustration of the meaning and effect. 
Describe a linear circuit in space, and a surface bounded by the 
circuit (Fig 6). Over this surface, which call S, let an impressed 
force V act normally, V being the same all over the surface. 
This system of force is irrotational everywhere except at the 
bounding circuit, where there is a circuital distribution of g 
the curl of e , of strength V. 

Now, our present proposition asserts that the disturbances 
due to V over the surface S are in every respect the same as 
those due to V (the same normal impressed force), spread 
over any other surface bounded by the same circuit. The 
comprehensiveness of this property may be illustrated by sup- 
posing that one surface is wholly in a non-conductor, whilst 


the other passes through a conductor ; or that one surface is 
wholly within a conductor, whilst the other passes out- of it 
and through another conductor insulated from the first. Thus, 
in the diagram, let S be the first surface (in section), and S t 
the second, passing through a conductor represented by the 
shaded region. The points A, B are where the common 
boundary of the two surfaces cuts the paper, whilst the arrows 
serve to show the direction of action of the impressed force. 
S 2 is another surface of impressed force, also reaching into the 
conductor. Now all these forcives (each by itself) will produce 
the same final state of displacement in the dielectric and 
electrification of the conductor, and will do so in the same 
manner; that is, the electromagnetic disturbances generated 
will be the same when expressed in terms of E and H. The 
distribution of energy, for example, and the stresses, will be 
the same. 

But there must be some difference made by thus shifting the 
source of energy. Obviously the nature of the flux of energy 
is changed. This being 

W = V(E-e )(H-h ), 

where we deduct the intrinsic forces to obtain the forces effec- 
tive in transferring energy, we see that every change made in 
the distribution of the intrinsic sources affects the flux of 
energy, in spite of the independence of E and H of their distri- 
butions (subject to the limitation mentioned). In our example, 
however, the only change is in the sheet of impressed force 

Production of Steady State due to Impressed Forcive by 
crossing of Electromagnetic Waves. Example of a Cir- 
cular Source. Distinction between Source of Energy 
and of Disturbance. 

89. It may be readily suspected from the preceding, that, 
as far as the production of electromagnetic disturbances goes, 
we may ignore e and h altogether, and regard the circuital 
vectors g and j as the real sources. This is, in fact, the case 
when ultimately analysed. In the example just taken the cir- 
cuit ABA is the source of the disturbances. That is, they 
emanate from this line. If disturbances were propagated 





infinitely rapidly there might be some difficulty in recognising 
the property, because the steady state appropriate to the 
instantaneous state of the impressed force would exist 
(if the conductor were away) ; but, as we shall see later, 
the speed of propagation is always finite, depending upon the 
values of c and /x- in the medium ; and conduction does not alter 
this property, although by its attenuating and distorting effects 
it may profoundly alter the nature of the resultant phenomena. 
With, then, a finite speed of propagation, we have merely to 
cause the impressed force to vary or fluctuate sufficiently 
rapidly to obtain distinct evidence of the emanation of waves 
from the real sources of disturbance. Thus, let the source be 

FIG. 7. 

circular in a plane perpendicular to the paper (Fig. 7), and A, B 
the points where it cuts the paper. When the source is suddenly 
started, the circle ABA is the first line of magnetic force. At 
any time t later, such that vt is less than the distance JAB, 
the electromagnetic disturbance will be confined wholly within 
a ring-shaped region having the circular source for core and of 
radius vt round A or round B. But when the distance vt in- 
creases to JAB overlapping commences, and a little later there 
is (as in the shaded part of the figure) a region occupied by two 
coincident waves crossing one another. Now, the union of 
these waves produces the steady state of displacement without 
magnetic force tha* e, within the shaded region the magnetic 


force vanishes, and the displacement is that which belongs to 
the final steady state. As time goes on, of course the shaded 
region enlarges itself indefinitely, although outside it is still a 
region occupied by electromagnetic disturbances going out to 
infinity. This supposes that there is no conductor in the 
way. Should there be one, then, as soon as it is struck by the 
initial electromagnetic wave, it becomes and continues to be 
a secondary source of disturbance, and the final steady state, 
different from the former, now arises from the superimposition 
of the primary waves and the secondary. 

It is, of course, impossible to go into detail at present, the 
object being merely to point out the difference between sources 
of energy and sources of disturbance. The source of energy 
only works when there is electric current at the spot, and this 
comes to it from the vortex lines (the circuital sources of dis- 
turbance). Thus, in the last figure, at time vt, when the ring 
of disturbance is of thickness 2vt, if the impressed force be in a 
circular disc whose trace is the straight line AB, the force is 
working in Aa and in B6, but inoperative elsewhere. Again, 
after overlapping has begun it is still working in Aac and B6c?, 
but is inoperative in cd, having done its work. Similarly 
when the sheet of impressed force has any other shape. The 
impressed force only works when it is allowed to work by the 
electromagnetic wave reaching it. 

To emphasize the matter, take another case. Let the 
impressed force in any telegraph circuit be confined to a single 
plane section across the conductor, so that the vortex line 
is a line on its surface, going round it. If this be at 
Valencia, we may shift the " seat of the E.M.F." to Newfound- 
land, provided we preserve continuity of connexion with the 
vortex line, as before explained; for instance, by extending 
the sheet over the whole surface of the conductor between the 
two places. The sheet at Newfoundland and the surface sheet 
will together produce the same effects as the original sheet at 
Valencia. Or, we may have the sheet entirely outside the 
circuit, provided only that it is bounded by the original vortex 
line, in touch with the conductor. 

What the practical interpretation of this extraordinary 
property is in connection with the "seat of E.M.F." of galvanic 
batteries and in electrolysis generally still remains obscure. 



All impressed voltages and gaussages are more or less difficult 
to understand. But there need be no doubt as to the general 
truth of the property if the circuitality of the current can be 

The Eruption of "47r"s. 

90. It may have been observed that the equations employed 
by me in the preceding differ from those in use in all mathe- 
matical treatises on the subject in a certain respect (amongst 
others), inasmuch as the constant 47r, which is usually so 
obtrusively prominent, has been conspicuous by its absence. 
This constant 4?r was formerly supposed to be an essential part 
of all electric and magnetic theories. One of the earliest 
results to which a student of the mathematics of electricity 
was introduced in pre-Maxwellian days was Coulomb's law of 
the relation between the density of the electric layer on a 
conductor and the intensity of electric force just outside it, say 


and, since this was proved by mathematics, it seemed that the- 
4?r was an essential ratio between two physical quantities, viz.^ 
electricity and the force it exerted on other electricity. Never 
a hint was given that this 4:r was purely conventional ; it was- 
not, indeed, even recognised to be conventional, and is not at 
the present day in some quarters. Then, again, at the begin- 
ning of magnetism, was Gauss's celebrated theorem proving 
mathematically that the total flux of magnetic force outward 
through a closed surface equalled precisely 4?r times the total' 
amount of magnetism enclosed within the surface ; and, for all 
that might appear to the contrary, this remarkable result 
flowed out necessarily from the properties of the potential 
function and its derivatives, and of the three direction cosine* 
of the normal to an element of the surface. It was funny 
very funny. How ever the 4?r managed to find its way in was- 
the puzzle in these and similar results ; for instance, in the 

, ...... (2) 

where //, is the permeability and K another physical property, 
the susceptibility to magnetisation of a substance. The dark 


mystery was carefully covered up by the mathematics. It was 
as hard to understand as the monarch found it to explain the 
presence of the apple in the dumpling how did the goodwife 
manage to get it in ? Nor was the matter rendered plainer by 
Maxwell's great treatise. Maxwell thought his theory of 
electric displacement explained the meaning of the 47r (in the 
corresponding electric theorem), as if it were a matter of 
physics, instead, as is the fact, of irrationally chosen units. 

As the present chapter has been mainly devoted to a general 
outline of electromagnetic theory expressed in formulae involv- 
ing rational units, it will be fitting, in bringing it to a conclu- 
sion, to explain here the relation these rational units bear to 
the ordinary irrational units. 

The Origin and Spread of the Eruption. 

91. The origin of the 4?r absurdity lay in the wisdom of 
our ancestors, literally. The inverse square law being recog- 
nised, say that two charges q l and q 2 repelled one another with 
a force varying inversely as the square of the distance between 
them; thus, 

...... (3) 

where a is a constant ; what was more natural than to make 
the expression of the law as simple as possible by giving the 
constant a the value unity, if, indeed, it were thought of at all ? 
Our ancestors could not see into the future, that is to say, 
beyond their noses, and perceive that this system would work 
out absurdly. They were sufficiently wise in their generation, 
and were not to blame. 

But, after learning that certain physical quantities bear to 
one another invariable relations, we should, in forming a syste- 
matic representation of the same, endeavour to avoid the intro- 
duction of arbitrary and unnecessary constants. This valuable 
principle was recognised to a small extent by our ancestors, as 
above ; it was emphasised by Maxwell and Jenkin in their 
little treatise on units in one of the Reports of the B. A. 
Committee on Electrical Standards (1863, Appendix C; p. 59 
of Spon's Reprint). Thus, referring to the magnetic law of 
inverse squares, we have the following : 


" The strength of the pole is necessarily defined as propor- 
tional to the force it is capable of exerting on any other pole. 
Hence the force / exerted between two poles of the strengths m 
and m v must be proportional to the product mm r The force 
f is also found to be inversely proportional to the square of the 
distance, D, separating the poles, and to depend on no other 
quantity ; hence, we have, unless an absurd and useless coefficient 
be introduced, 

f=mmJD*. ..... ..... (4) 

Observe the words which I have italicised. When it is con- 
sidered what Maxwell had then done in the way of framing a 
broad theory of electromagnetism, it is marvellous that he 
should have written in that way. By mere force of habit one 
might, indeed, not consider there to be anything anomalous 
about the 4?r in Coulomb's and Gauss's theorems. But did not 
Maxwell's electrostatic energy K^ 2 /8?r and his magnetic energy 
pffi/Sir per unit volume loudly proclaim that there was some- 
thing radically wrong in the system to lead to such a mode of 
expression, which fault should be attended to and set right ab 
initio, especially in framing a permanent system of practical 
units, which was what Maxwell and his colleagues were about ? 
It would seem that the proclamation fell upon deaf ears, for 
not only were the units irrationally constructed, but in his 
Treatise, which followed some years later, we find the following 
statement (p. 155, second edition). After an account of his 
theory of electric displacement, we are told that " the theory 
completely accounts for the theorem of Art. 77, that the total 
induction through a closed surface is equal to the total quantity 
of electricity within the surface multiplied by 4?r. For what 
we have called the induction through the surface is simply the 
electric displacement multiplied by 47r, and the total displace- 
ment outward is necessarily equal to the total charge within 
the surface." That is, his theory of electric displacement 
accounts for the 47r. So it seems to do; and yet the 4?r has no 
essential connection with his theory, or with any one else's. 
Though by no means evident until it is pointed out, it is entirely 
a question of the proper choice of units, and is independent of 
all theories of electricity. It depends upon something much 
more fundamental. 


The Cure of the Disease by Proper Measure of the Strength 
of Sources. 

92. When looked into carefully, the question is simply 
this : What is the natural measure of the strength of a source ? 
Suppose, for instance, we have a source of heat in a medium 
which does not absorb heat, how should we measure the 
intensity of the source ? Plainly by the amount of heat emitted 
per second, passing out through any surface enclosing the 
source. If the flux of heat be isotropically regular, its density 
will vary inversely as the square of the distance from a point 
source, giving 

...... (5) 

if C be the heat flux (per unit area) at distance r from the 
source of strength S. If we knock out the 4?r we shall 
obviously have an unnatural measure of the strength of the 

Similarly, if we send water along a pipe and let it flow from 
its end, which may therefore be regarded as a source, we 
should naturally measure its strength in a similar manner, viz., 
by the current in the pipe, or by the total outflow. 

Now in an electric field, or in a magnetic field, or in the 
field of any vector magnitude, we have everywhere mathe- 
matically analogous cases. We find, for instance, that electro- 
static force is distributed like velocity in an incompressible 
liquid, except at certain places, where it is, by analogy, 
generated, or has its source. If, then, we observe that the 
flux of force through a closed surface is not zero, there must 
be sources within the region enclosed, and the natural measure 
of the total strength of the sources is the total flux of force 
itself. I have put this in terms of electric force rather than of 
electric displacement, merely to exemplify that the matter has 
no particular connexion with electric displacement. In the 
former case it is a source of "electric force" that is considered; 
in the latter, it would be of displacement ; and the principle 
concerned in a rational reckoning of the strength of a source is 
the same in either case. It is a part of the fitness of things, 
and holds good in the abstract theory of the space-variation of 
vector magnitudes, apart from all physical application. 


Using, temporarily, the language of lines of force, or of tubes 
of force (which, however, as here, sometimes works out rather 
nonsensically), we may say that a unit pole sends out one line 
of force, or one tube, when rationally estimated. 

Next there is the proper measure of the strength of cir- 
cuital fluxes to be considered; electric current, for instance. 
The universal property here is that the circulation of the 
magnetic force is proportional to the current through the 
circuit ; and the natural pleasure of the strength of the current 
is the circuitation itself, without, as usual, dividing by 4n-. 
Now this division by 4?r arises out of the irrational reckoning 
of the strength of point sources. We may therefore expect 
that when the point sources are measured rationally, the 4r 
will disappear from the reckoning of electric current, making 
the circuitation of magnetic force the proper reckoning. This 
is so, as may be easily seen by substituting for a linear 
electric current an equivalent magnetic shell, and so bringing 
in point sources distributed over its two faces. 

Thus, in rational units, if we have a point source q of dis- 
placement, and a point source m of induction, we have 

D = ?/47rr 2 , B = m/47rr 2 , . . . (6) 

to express the displacement and induction at distance r, when 
the fluxes emanate isotropically ; and 

. . (7) 

if q/c and m/p are the measures of the sources of electric and 
magnetic force respectively. In the magnetic case m may repre- 
sent the strength of a pole, on the understanding that (since 
the induction is really circuital) we ignore the flux coming to the 
pole (as along a filamentary magnet), and consider only the 
diverging induction. Also, 

H = C/27rr ...... (8) 

expresses the intensity of magnetic force at distance r from a 
long solitary straight current of strength C. 

Obnoxious Effects of the Eruption. 

93. Considering merely the formulae belonging to point 
sources with uniform divergence, we see that the effect of 


changing from irrational to rational units is to introduce 4?r. 
If this were all, we might overlook the fundamental irration- 
ality and use irrational units for practical convenience. But, 
as a matter of fact, it works out quite differently. For the 
unnatural suppression of the 4:r in the formulae of central 
force, where it has a right to be, drives it into the blood, there 
to multiply itself, and afterwards break out all over the body 
of electromagnetic theory. The few formulae where 4?r should 
be are principally scholastic formulae and little used; the many 
formulae where it is forced out are, on the contrary, useful 
formulae of actual practice, and of the practice of theory. A 
practical theorist would knock them out merely from the 
trouble they give, let alone the desire to see things in their 
right places. Furthermore, it should be remarked that the 
irrationality of the formulae is a great impediment in the way 
of a clear understanding of electromagnetic theory. The 
interpretation of equation (2) above, for instance, or of the 
similar well-known equation 

$ = ) + 47r|ji, 

presents some difficulty even to a student of ability, unless he 
be given beforehand a hint or two to assist him. For if K is 
a rational physical quantity, then u. cannot be ; or if /A is right, 
then K must be wrong. Or ^, ^, and ^L must be incongruous. 
The 4?r is also particularly inconvenient in descriptive matter 
relating to tubes of force or flux, and in everything connected 
with them. 

This difficulty in the way of understanding the inner mean- 
ing of theory is still further increased by the 4n- not entering 
into the magnetic formulae in the same way as into the electric. 
Thus, it is jp and ^/4w which are analogues. Again, in many 
of the irrational formulae the irrationality appears to disappear. 
For instance, in J^/o, in J31C/, in J($5 * n ^j an( i i n some 
others. This is because both the factors are irrational, and the 
two irrationalities cancel. It looks as if J) were the flux 
belonging to ($, but it is not. In reality, we have, if x = (47r)*, 

therefore, ED = 

whilst c is the same in both irrational and rational units. 


A Plea for the Removal of the Eruption by the Radical Cure. 

94. The question now is, what is to be done? Are we 
modern pigmies, who by looking over the shoulders of the 
giants can see somewhat further than they did, to go on perpe- 
trating and perpetuating their errors for ever and ever, and 
even legalising them ? If they are to be enforced, it is to b& 
hoped that it will not be made a penal offence not to use the 
legal and imperial units. 

The " brain-wasting perversity " of the British nation in sub- 
mitting year after year to be ruled by such a heterogeneous 
and incongruous collection of units as the yard, foot, inch, mile,, 
knot, pound, ounce, pint, quart, gallon, acre, pole, horse-power, 
etc., etc., has been repeatedly lamented by would-be reformers, 
who would introduce the common-sense decimal system ; and 
amongst them have been prominent electricians who hoped to- 
insert the thin end of the wedge by means of the decimal sub- 
division of the electrical units, and their connection with the 
metre and gramme, and thus lead to the abolition of the present 
British system of weights and measures with its absurd and 
useless arbitrary connecting constants. But what a satire it 
is upon their labours that they should have fallen into the very 
pit they were professedly avoiding ! The perverse British 
nation practically the British engineers have surely a right 
to expect that the electricians will first set their own house in 

The ohm and the volt, etc., are now legalised, so that, as I 
am informed, it is too late to alter them. This is a non seq., 
however, for the yard and the gallon are legalised ; and if it is 
not too late to alter them, it cannot be too late to put the new- 
fangled ohm and its companions right. It is never too late to 
mend. No new physical laboratory determinations will be 
needed. The value of TT has been calculated to- hundreds of 
places of decimals ; so that rational ohms, volts, etc., can be now 
fixed with the same degree of accuracy as the irrational ones, 
by any calculator. 

When I first brought up this matter in The Electrician ia 
1882-3, explaining the origin of the 4?r absurdity and its cure, 
I did not go further than to use rational units in explaining the 


theory of potentials, scalar and vector, and similar matters; then 
returning to irrational units in order to preserve harmony with 
the formulae in Maxwell's treatise, and I did not think a change 
was practicable, on account of the B. A. Committee's work, and 
the general ignorance and want of interest in the subject. But 
much has happened since then. The spread of electrical know- 
ledge has been immense, concurrent with the development of 
electrical industries, to say nothing of theoretical and experi- 
mental developments. I therefore now think the change is 
perfectly practicable. At any rate, some one must set the 
example, if the change is to occur. I have, therefore, in the 
preceding, wholly avoided the irrational units ab initio ; and 
shall continue to use rational units in the remainder of this 

So far as theoretical papers and treatises are concerned, 
there is no difficulty. Every treatise on Electricity should be 
done in rational formulae, their connexion with the irrational 
(so long as they exist) being explained separately (in a chapter 
at the end of the book, for instance), along with the method of 
converting into volts, amperes, etc., the present practical units. 
At present you have to first settle whether to use the electro- 
static or the electromagnetic units, and then introduce the 
appropriate powers of 10. If rational formula) are used, then, 
in addition, you must first insert the constant 4?r in certain 
places, so long as the irrational units last. 

When, however, the real advantages of the rational system 
become widely recognized and thoroughly assimilated, then 
will come a demand for the rationalisation of the practical 
units. Even at present the poor practician is complaining 
that he cannot even pass from magnetomotive force to am- 
pere-turns without a " stupid " 4?r coefficient getting in the 
way. That the practical electrical units should be reformed 
as a preliminary to the general reform of the British units 
requires no argument to maintain. That this general reform is 
coming I have not the least doubt. Even the perversity of the 
British engineer has its limits. 

Rational v. Irrational Electric Poles. 

95. We may now briefly compare some of the more impor- 
tant formulae in the two systems. Let us denote quantities in 


rational units as in the preceding (except that when vector rela- 
tions are not in question we need not employ special type), and 
the corresponding quantities in irrational units by the same 
symbols with the suffix f ; thus, q and &. Also denote (47r) J by x. 
Let q i be the irrational charge which repels an equal 
irrational charge at distance r with the same force F as a 
rational charge q repels an equal clyirge at the same distance. 1 

.... (9) 


2 = *% ....... (10) 

As regards the ratio of the units, it is sufficient merely to 
observe that the magnitude of the number expressing a 
quantity varies inversely as the size of the unit. This 
applies throughout, so that we need not bring in units at all, 
but keep to the concretes appearing in formulae. 
By (10) we shall have 

if p is volume-density of electrification, a- surface density, 
D displacement, C electric current density (or else the total 
displacement and current). 

Since D = cE, and D^cEJiTr, .... (12) 
whilst D = #Di by (11), we have also 

_E< A, V, e t P, x 13 x 


where A is the time-integral of E, V the line-integral of E, or 
voltage, e impressed electric force, P electric potential or poten- 
tial difference. 

The permittivity c is the same in both systems. So is the 
ratio c/c of the permittivity to that of ether, or the specific 
inductive capacity (electric). The rational permittance of a 
unit cube condenser is c, and the irrational is cjkir. If S is 
the permittance of any condenser 

S-^S* ...... (14) 


and its energy is 

. . . (15) 
if Q is its charge and V its voltage. We also have 

JED = JE i D i = JcE2 = icE t 2/47r, . . . (16) 


the first and second sets relating to electric energy, the third 
to magnetic energy. 
By Ohm's law 

V = RC, and V = RC,; . . (19) 

therefore by (11) and (13) 

* 2 =l=-^> ..... (20) 
where r is resistivity, so that 

. . (21) 

Rational v. Irrational Magnetic Poles. 

96. If the repulsion between two magnetic poles m and m 
is F at distance r, and this is also the repulsion between imu- 
tional poles m i and m i5 we have 


where /x. is the inductivity of the medium, common to the two 
systems, as is likewise /A//A O the ratio of p. to that of the ether, 
or the permeability. So 

m = xm i ....... (23) 

Here some discrepancies come in. For 

B = /*H, and 3, = ^; . . (24) 

, Tfl Hi Bv l2v lit /ner\ 

so we have *=_ = _! = _!= - ^ . . . (25) 

where ft is magnetic potential or gaussage. If I be intrinsic 
magnetisation (intensity of), then 

and l^phjtf, . . . (26) 


from which, and by (25), we have 

I = *Ii , - - (27) 

The equation B = /*H, in the sense used by me, expands to 
B = /x(h + F), (28) 

where h is intrinsic, and F is the magnetic force of the field. 

^ = ^ (1+K) = ^(1+4,), . . . (29) 

.0 we have, by (28) and (26), 

B = I + fi F + /v<F, (30) 

where /* KF is the induced magnetisation. 

If L is inductance, ^LC^L^ 2 , whence *=* = J^ if 

L M 

M is mutual inductance. 
In the common equation, 

B i = F i + 47rI i (31), 

the intrinsic and induced magnetisations are lumped together 
for one thing, and it is assumed that fi = l. The quantity K 
is thus essentially a numeric, whilst /* is only a numeric by 
assumption. But whilst p = i*>i, we have K = x*K i . Although 
magnetisation, whether intrinsic or induced, are of the same 
kind as induction, yet the reckoning is discrepant. Compare 
(27) with (25) for B and B f . In (27) also, I may be either 
intrinsic or induced, so far as the ratio x goes. 
The common equation, div "D i = p { , becomes 

divD=/>, ...... (32) 

and the characteristic equation of Poisson becomes 

V2 P= _ /) / c , (33) 

the 4n- going out by the rationalisation. But P itself is given 

by P = 2/>/47iTC (34) 

The definition of current density, 

curl H f = 471-0^, 
becomes, curlH = C (35) 


Other changes readily follow. But now that I have explicitly 
stated the relation between my rational formulae and the 
ordinary, I leave the irrationals for good, I hope and return 
to the rational and simplified formulae, which are so much 



According to Maxwell's theory of electric displacement, 
disturbances in the electric displacement and magnetic induc- 
tion are propagated in a non-conducting dielectric after the 
manner of motions in an incompressible solid. The subject is 
somewhat obscured in Maxwell's treatise by his equations of 
propagation containing A, , J, all of which are functions 
considerably remote from the vectors which represent the state 
of the field, viz., the electric and magnetic forces, and by some 
dubious reasoning concerning M* and J. There is, however, no 
doubt about the statement with which I commenced, as it 
becomes immediately evident when we ignore the potentials 
and use E or H instead, the electric or the magnetic force. 

The analogy has been made use of in more ways than one, 
and can be used in very many ways. The easiest of all is to 
assume that the magnetic force is the velocity of the medium, 
magnetic induction the momentum, and so on, as is done by 
Prof. Lodge (Appendix to "Modern Views of Electricity"). 
I have also used this method for private purposes, on account 
of the facility with which electromagnetic problems may be 
made elastic-solid problems. I have shown that when impressed 
electric force acts it is the curl or rotation of the electric force 
which is to be considered as the source of the resulting distur- 
bances. Now, on the assumption that the magnetic force is 
the velocity in the elastic solid, we find that the curl of the 
impressed electric force is represented simply by impressed 
mechanical force of the ordinary Newtonian type. This is 
very convenient. 


But the difficulties in the way of a complete and satisfactory 
representation of electromagnetic phenomena by an elastic- 
solid ether are insuperable. Recognising this, Sir W. Thomson 
has recently brought out a new ether ; a rotational ether. It 
is incompressible, and has no true rigidity, but possesses a quasi' 
rigidity arising from elastic resistance to absolute rotation. 

The stress consists partly of a hydrostatic pressure (which I 
shall ignore later), but there is no distorting stress, and its 
place is taken by a rotating stress. It gives rise to a trans- 
lational force and a torque. If E be the torque, the stress on 
any plane N (unit normal) is simply VEN, the vector product 
of the torque and the normal vector. 

The force is curl E. We have therefore the equation of 

- curl E = juH, 

if H is the velocity and p the density. But, alas, the torque 
is proportional to the rotation. This gives 

curl H = cE, 

where c is the compliancy, the reciprocal of the quasi-rigid ity. 

Now these are the equations connecting electric and magnetic 
force in a non-conducting dielectric, when p, is the inductivity 
and c the permittancy. We have a parallelism in detail, not 
merely in some particulars. The kinetic energy JftH 2 repre- 
sents the magnetic energy, and the potential energy icE 2 the 
electric energy. The vector-flux of energy is VEH, the activity 
of the stress. 

This mode of representation differs from that of Sir W. 
Thomson, who represents magnetic force by rotation. This 
system makes electric energy kinetic, and magnetic energy 
potential, which I do not find so easy to follow. 

Now let us, if possible, extend our analogy to conductors. 
Let the translational and the rotational motions be both fric- 
tionally resisted, and let the above equations become 

-curl E= 

where g is the translational frictionality ; Tc will be considered 
later. We have now the equations of electric and magnetic 
force in a dielectric with duplex conductivity, k being the 


electric and g the magnetic conductivity (by analogy with elec- 
tric force, hut a frictionality in our present dynamical analogy). 

We have, therefore, still a parallelism in every detail. We 
have waste of energy by friction <?H 2 (translatibnal) and &E 2 
(rotational). If g/fj> = k/c the propagation of disturbances 
will take place precisely as in a non-conducting dielectric, 
though with attenuation caused by the loss of energy. 

To show how this analogy works out in practice, consider a 
telegraph circuit, which is most simply taken to be three 
co-axial tubes. Let, A, B, and C be the tubes ; A the inner- 
most, C the outermost, B between them; all closely fitted. 
Let their material be the rotational ether. In the first place, 
suppose that there is perfect slip between B and its neighbours. 
Then, when a torque is applied to the end of B (the axis of 
torque to be that of the tubes), and circular motion thus 
given to B, the motion is (in virtue of the perfect slip) trans- 
mitted along B, without change of type, at constant speed, and 
without affecting A and C. 

This is the analogue of a concentric cable, if the conductors 
A and C be perfect conductors, and the dielectric B a perfect 
insulator. The terminal torque corresponds to the impressed 
voltage. It should be so distributed over the end of B that the 
applied force there is circular tangential traction, varying 
inversely as the distance from the axis ; like the distribution 
of magnetic force, in fact. 

Now, if we introduce translational and rotational resistance 
in B, in the above manner, still keeping the slip perfect, we 
make the dielectric not only conducting electrically but also 
magnetically. This will not do. Abolish the translational re- 
sistance in B altogether, and let there be no slip at all between 
B and A, and B and C. Let also there be rotational resistance 
in A and C. 

We have now the analogue of a real cable : two conductors 
separated by a third. All are dielectrics, but the middle one 
should have practically very slight conductivity, so that it is 
pre-eminently a dielectric ; whilst the other two should have 
very high conductivity, so that they are pre-eminently con 
ductors. The three constants, ft, c, k, may have any value iii 
the three tubes, but practically k should be in the middle tubs 
a very small fraction of what it is in the others. 



It is remarkable that the quasi-rotational resistance in A and 
C should tend to counteract the distorting effect on waves of 
the quasi-rotational resistance in B. But the two rotations, 
it should be observed, are practically perpendicular, being 
axial or longitudinal (now) in A and C, and transverse or 
radial in B ; due to the relative smallness of Tc in the middle 

To make this neutralising property work exactly we must 
transfer the resistance in the tubes A and C to the tube B, at 
the same time making it translational resistance. Also restore 
the slip. Then we can have perfect annihilation of distortion 
in the propagation of disturbances, viz., when k and g are so 
proportioned as to make the two wastes of energy equal. In 
the passage of a disturbance along B there is partial absorption, 
but no reflection. 

But as regards the meaning of the above k there is a diffi- 
culty. In the original rotational ether the torque varies as the 
rotation. If we superadd a real frictional resistance to rotation 
we get an equation of the form 

E being (as before) the torque, and H the velocity. But this 
is not of the right form, which is (as above) 


therefore some special arrangement is required (to produce the 
dissipation of energy &E 2 ), which does not obviously present 
itself in the mechanics of the rotational ether. 

On the other hand, if we follow up the other system, in 
which magnetic force is allied with rotation, we may put g = 0, 
let - E be the velocity and H the torque ; /* the compliancy, 
c the density, and k the translational frictionality.. This gives 

curl H = &E + cE. 

We thus represent a homogeneous conducting dielectric, with 
a translational resistance to cause the Joulean waste of energy e 


But it is now seemingly impossible to properly satisfy the con- 
ditions of continuity at the interface of different media. For 
instance, the velocity -E should be continuous, .but we do not 
have normal continuity of electric force at an interface. In 
the case of the tubes we avoided this difficulty by having the 
velocity tangential. 

Either way, then, the matter is left, for the present, in an 
imperfect state. 

In the general case, the d/dt of our equations should receive 
an extended meaning, on account of the translational motion 
of the medium. The analogy will, therefore, work out less 
satisfactorily. And it must be remembered that it is only an 
analogy in virtue of similitude of relations. We cannot, for 
instance, deduce the Maxwellian stresses and mechanical forces 
on charged or currented bodies. The similitude does not ex- 
tend so far. But certainly the new ether gees somewhat 
further than anything known to me that has been yet proposed 
in the way of a stressed solid. 

[P.S. The special reckonings of torque and rotation in the 
above are merely designed to facilitate the elastic-solid and 
electromagnetic comparisons without unnecessary constants.] 



Scalars and Vectors. 

97. Ordinary algebra, as is well known, treats of quantities 
and their relations. If, however, we examine geometry, we 
shall soon find that the fundamental entity concerned, namely 
a straight line, when regarded as an entity, cannot be treated 
simply as a quantity in the algebraical sense. It has, indeed, 
size, viz., its length ; but with this is conjoined another impor- 
tant property, its direction. Taken as a whole, it is a Vector. 
In contrast with this, an ordinary quantity, having size only, is- 
a Scalar. 

Again, if we consider the mathematics of physical questions, 
we find two distinct kinds of magnitudes prominently present. 
All such magnitudes as mass, density, energy, temperature, are 
evidently quantities in the simple algebraical sense; that is, 
scalar magnitudes, or simply scalars. A certain (it may be 
an unstated) unit of density, for instance, being implied, any 
density is expressed by a simple number. (The question of the 
"dimensions " of physical magnitudes is not in question.) All 
magnitudes whatever which have no directional peculiarity 
and which are each specified by a single number are scalars, 
and subject to scalar algebra. 

But such magnitudes as displacement, velocity, acceleration^ 
force, momentum, electric current, &c., which have direction as 
well as size, and which are fully specified by statement of the 
size and direction, are vector magnitudes, or simply vectors. 


Now, just as there is an algebra and analysis for scalars, so is 
there a vector algebra and analysis appropriate to vectors; and it 
is the object of the present chapter to give a brief account of 
the latter, especially in respect to its application to electromag- 

Characteristics of Cartesian and Vectorial Analysis. 

98. Algebraical or analytical geometry in the usual Cartesian 
form, though dealing ultimately with vectors, is not vectorial 
algebra. It is, in fact, a reduction to scalar algebra by resolu- 
tion of every vector into three rectangular components, which 
are manipulated as scalars. Similarly, in the usual treatment 
of physical vectors, there is an avoidance of the vectors them- 
selves by their resolution into components. That this is a 
highly artificial process is obvious, but it is often convenient. 
More often, however, the Cartesian mathematics is ill-adapted 
to the work it has to do, being lengthy and cumbrous, and 
frequently calculated to conceal rather than to furnish and 
exhibit useful results and relations in a ready manner. When 
we work directly with vectors, we have our attention fixed 
upon them, and on their mutual relations ; and these are 
usually exhibited in a neat, compact, and expressive form, 
whose inner meaning is evident at a glance to the practised 
eye. Put the same formula, however, into the Cartesian form, 
and what a difference ! The formula which was expressed by 
a few letters and symbols in a single line, readable at once, 
sometimes swells out and covers a whole page ! A very close 
study of the complex array of symbols is then required to find out 
what it means ; and, e v en though the notation be thoroughly 
symmetrical, it becomes a work of time and great patience. 
In this interpretation we shall, either consciously or uncon- 
sciously, be endeavouring to translate the Cartesian formulae 
into the language of vectors. 

Again, in the Cartesian method, we are led away from the 
physical relations that it is so desirable to bear in mind, to the 
working out of mathematical exercises upon the components. 
It becomes, or tends to become, blind mathematics. It was 
once told as a good joke upon a mathematician that the poor 
man went mad and mistook his symbols for realities ; as M for 
the moon and S for the sun. There is another side to the 


story, however. If our object be ultimately physical, rathei 
than mathematical, then the more closely we can identify the 
symbols with their physical representatives the more usefully 
can we work, with avoidance of useless though equally true 
mathematical exercises. The mere sight of the arrangement 
of symbols should call up an immediate picture of the physics 
symbolised, so that our formulae may become alive, as it were. 
Now this is possible, and indeed, comparatively easy, in 
vectorial analysis ; but is very difficult in Cartesian analysis, 
beyond a certain point, owing to the geometrically progressive 
complexity of the expressions to be interpreted and manipu- 
lated. Vectorial algebra is the natural language of vectors, 
and no one who has ever learnt it (not too late in life, how- 
ever) will ever care to go back from the vitality of vectors to 
the bulky inanimateness of the Cartesian system. 

Abstrusity of Quaternions and Comparative Simplicity 
gained by ignoring them. 

99. But supposing, as is generally supposed, vector 
algebra is something "awfully difficult," involving meta- 
physical considerations of an abstruse nature, only to be 
thoroughly understood by consummately profound metaphysico- 
mathematicians, such as Prof. Tait, for example. Well, if so, 
there would not be the slightest chance for vector algebra and 
analysis to ever become generally useful ; and I should not be 
writing this, nor should I have, for several years past, persisted 
in using vector algebra in electromagnetic theory a prophet 
howling in the wilderness. It will readily be concluded, then, 
that I believe that the vector analysis is going to become 
generally used in scientific work, and that what is needed is 
not "awfully difficult." There was a time, indeed, when I, 
although recognising the appropriateness of vector analysis in 
electromagnetic theory (and in mathematical physics gene- 
rally), did think it was harder to understand and to work 
than the Cartesian analysis. But that was before I had 
thrown off the quaternionic old-man-of-the-sea who fastened him- 
self on my shoulders when reading the only accessible treatise 
on the subjects-Prof. Tait's Quaternions. But I came later 
to see that, so far as the vector analysis I required was 
concerned, the quaternion was not only not required, but was 


a. positive evil of no inconsiderable magnitude ; and that by its 
avoidance the establishment of vector analysis was made quite 
simple and its working also simplified, and that it could be con- 
veniently harmonised with ordinary Cartesian work. There is 
not a ghost of a quaternion in any of my papers (except in 
one, for a special purpose). The vector analysis I use may be 
described either as a convenient and systematic abbreviation of 
Cartesian analysis ; or else, as Quaternions without the quater- 
nions, and with a simplified notation harmonising with Car- 
tesians. In this form, it is not more difficult, but easier to 
work than Cartesians. Of course you must learn how to work 
it. Initially, unfamiliarity may make it difficult. But no 
amount of familiarity will make Quaternions an easy subject. 

Maxwell, in his great treatise on Electricity and Magnetism, 
whilst pointing out the suitability of vectorial methods to the 
treatment of his subject, did not go any further than to freely 
make use of the idea of a vector, in the first place, and to 
occasionally express his results in vectorial form. In this way 
his readers became familiarised with the idea of a vector, and 
also with the appearance of certain formula) when exhibited in 
the quaternionic notation. They did not, however, derive any 
information how to work vectors. On the whole, I am inclined 
to think that the omission of this information has not tended 
to impede the diffusion of a knowledge of vector analysis. 
For, had he given an account of the theory, he would certainly 
have followed the Hamilton-Tait system; and this would 
probably, for reasons I shall shortly mention, have violently 
prejudiced his readers against the whole thing. 

But the diffusion of vector analysis has, undoubtedly, in 
my opinion, been impeded by the absence of sufficiently ele- 
mentary works on the subject, with a method of establishment 
of principles adapted to ordinary minds, and with a con- 
veniently workable notation. For the reader of Maxwell's 
treatise who desired to learn to work vectors in analysis had 
either feo go to Hamilton's ponderous volumes, or else to Prof. 
Tait's treatise. The former are out of the question for initiatory 
purposes. But the latter is excessively difficult, although de- 
scribed as " an elementary treatise " not the same thing as 
" a treatise on the elements." The difficulty arises in a great) 
measure from the quaternionic basis. 


Elementary Vector Analysis independent of the 

100. Suppose a sufficiently competent mathematician 
desired to find out from the Cartesian mathematics what vector 
algebra was like, and its laws. He could do so by careful 
inspection and comparison of the Cartesian formula). He would 
find certain combinations of symbols and quantities occurring 
again and again, usually in systems of threes. He might 
introduce tentatively an abbreviated notation for these com- 
binations. After a little practice he would perceive the laws 
according to which these combinations arose and how they 
operated. Finally, he would come to a very compact system 
in which vectors themselves and certain simple functions of 
vectors appeared, and would be delighted to find that the rules 
for the multiplication and general manipulation of these vectors 
were, considering the complexity of the Cartesian mathematics 
out of which he had discovered them, of an almost incredible 
simplicity. But there would be no sign of a quaternion in his 
results, for one thing ; and, for another, there would be no 
metaphysics or abstruse reasoning required to establish the 
rules of manipulation of his vectors. Vector analysis is, in its 
elements, entirely independent of the exceedingly difficult 
theory of quaternions ; that is, when the latter is treated 
quaternionically ab initio. 

" Quaternion " was, I think, defined by an American school- 
girl to be " an ancient religious ceremony." This was, however, 
a complete mistake. The ancients unlike Prof. Tait knew 
not, and did not worship Quaternions. The quaternion and 
its laws were discovered by that extraordinary genius Sir W. 
Hamilton. A quaternion is neither a scalar, nor a vector, but 
a sort of combination of both. It has no physical representa- 
tives, but is a highly abstract mathematical concept." It is the 
" operator " which turns one vector into another. It has a 
stretching faculty first, to make the one vector become as long 
as the other ; and a rotating faculty, to bring the one into 
parallelism with the other. 

Now in Quaternions the quaternion is the master, and lays 
down the law to the vector and scalar. Everything revolves 


round the quaternion. The laws of vector algebra themselves 
^re_established through quaternions, assisted by the imaginary 
N/^l. But I am not sure that any one has ever quite under- 
-stood this establishment. It is done in the second chapter of 
Tait's treatise. I never understood it, but had to pass on. 
That the establishment is not demonstrative may be the reason 
of the important changes made therein in the third edition. 
But it is still undemonstrative to me, though much improved. 
Now this relates to the very elements of the subject, viz., the 
scalar and vector products of a pair of vectors, the laws of 
which are quite plain in the Cartesian mathematics. Clearly, 
then, the quaternionic is an undesirable way of beginning the 
subject, and impedes the diffusion of vectorial analysis in a way 
which is as vexatious and brain-wasting as it is unnecessary. 

Tait v. Gibbs and Gibbs v. Tait. 

101. Considering the obligations I am personally under to 
Prof. Tait (in spite of that lamentable second chapter), it does 
seem ungrateful that I should complain. But I have at heart 
the spread of a working knowledge of elementary vector 
analysis quite as much as Prof. Tait has the extension of the 
theory of quaternions. Besides, Prof. Tait has assumed a very 
conservative attitude in relation to Hamilton's grand system. 
For instance, to "more than one correspondent" who had 
written for explanation of something they found obscure and 
the same thing occurred to me described in his treatise by 
"It is evident, . . .", he, "on full consideration," decides not 
to modify it, but to italicise the words ! He also told them 
that if they did not see it, in the light of certain preceding 
parts of the treatise, then they had "begun the study of 
Quaternions too soon" (Third edition, p. 110). This is as 
characteristic of the sardonic philosopher as a certain heavy 
kind of " flippancy " is of the Cockney. Again, in his Preface 
he states one cause of the little progress made in the develop- 
ment of Quaternions to be that workers have (especially in 
France) been more intent on modifying the notation or the 
mode of presentation of the elementary principles, than in 
extending the application of the calculus. "Even Prof. 
Willard Gibbs must be ranked as one of the retarders of 
quaternionic progress, in virtue of his pamphlet on Vector 


Analysis, a sort of hermaphrodite monster, compounded of the 
notations of Hamilton and Grassmann." Grassmann, I may 
observe, established, inter alia, a calculus of vectors, but not 
of quaternions. 

Prof. W. Gibbs is well able to take care of himself. I may, 
however, remark that the modifications referred to are evidence 
of modifications being felt to be needed ; and that Prof. Gibbs's 
pamphlet (NOT PUBLISHED, Newhaven, 1881-4, pp. 83), is not 
a quaternionic treatise, but an able and in some respects 
original little treatise on vector analysis, though too condensed 
and also too advanced for learners' use ; and that Prof. Gibbs, 
being no doubt a little touched by Prof. Tait's condemnation, 
has recently (in the pages of Nature) made a powerful defence of 
his position. He has by a long way the best of the argument, 
unless Prof. Tait's rejoinder has still to appear. Prof. Gibbs 
clearly separates the quaternionic question from the question 
of a suitable notation, and argues strongly against the 
quaternionic establishment of vector analysis. I am able (and 
am happy) to express a general concurrence of opinion with 
him about the quaternion, and its comparative uselessness in 
practical vector analysis. As regards his notation,' however, I 
do not like it. Mine is Tait's, but simplified, and made to 
harmonise with Cartesians. 

Abolition of the Minus Sign of Quaternions. 

102. In Quaternions, the square of a unit vector is 1. 
This singular convention is quaternionically convenient. But in 
the practical vector analysis of physics it is particularly incon- 
venient, being indeed, an obtrusive stumbling-block. All 
positive scalar products have the minus sign prefixed ; there is 
thus a want of harmony with scalar investigations, and a 
difficulty in readily passing from Cartesians to Vectors and 
conversely. My notation, on the other hand, is expressly 
arranged to facilitate this mutual conversion. 

As regards the establishment of the elementary vector algebra, 
that is quite simple (freed from the quaternion) ; it all follows 
from the definitions of a vector and of the scalar and vector 
products of a pair of vectors. 

Now I can imagine a quaternionist (unless prejudiced) admit- 
ting the simplicity of establishment and of operation, and the 


convenience of the notation, and its sufficiency for practical re- 
quirements up to a certain point ; and yet adding the inquiry 
whether there is not, over and above this vector analysis, a theory 
of Quaternions which is overlooked. To this I would reply, Cer- 
tainly, but it is not food for the average mathematician, and 
can, therefore, never be generally used by him, his practical 
requirements being more suitably satisfied by the rudimentary 
vector analysis divested of the mysterious quaternion. This 
does not exclude the important theory of y and its applications. 
Prof. Gibbs would, I think, go further, and maintain that the 
anti- or ex-quaternionic vectorial analysis was far superior to 
the quaternionic, which is^ uniquely adapted to three dimen- 
sions, whilst the other admits of appropriate extension to more 
generalised cases. I, however, find it sufficient to take my 
stand upon the superior simplicity and practical utility of the 
ex-quaternionic system.. 

We may, however, if we wish to go further, after ex- 
quaternionic establishment of vector algebra, conjoin the scalar 
and vector, and make the quaternion, and so deduce the whole 
body of Quaternions. But sufficient for the day is the labour 
thereof ; and we shall* now be concerned with scalars and 
vectors only. The reader should entirely divest his mind of 
any idea that we are concerned with the imaginary ^^ "1 in 
vector analysis. Also, he should remember that unfamiliarity 
with notation and processes may give an appearance of difficulty 
that is entirely fictitious, even to an intrinsically easy matter ; 
so that it is necessary to thoroughly master the notation and 
ideas involved. The best plan is to sit down and work ; all 
that books can do is to show the way. 

Type for Vectors. Greek, German, and Roman Letters un- 
suitable. Clarendon Type suitable. Typographical 
Backsliding in the Present Generation. 

103. We should, in the first place, fix how to represent 
vectors, although in reality this is the outcome of experience. 
A vector may obviously be denoted by a single letter ; and, 
having defined certain letters to stand for scalars and others 
for vectors, it is certainly unnecessary that the vectors should 
be distinguished from the scalars by the use of different kinds 
of types. But practical experience shows that it is very 


desirable that this should be done, in order to facilitate the 
reading of vectorial work, by showing at a glance which letters 
are vectors and which are scalars, and thus easing the stress 
and strain on the memory. This is all the more important 
because the manipulation of vectors sometimes differs from that 
of scalars. 

Now Prof. Tait usually indicates vectors by Greek letters. 
But it is well known that a considerable familiarity with the 
Greek letters such as is acquired by studying the literature 
of ancient Greece is required before they can be read and 
manipulated with facility. On the other hand, few are Greek 
scholars, and in fact many people think it is about time the 
dead languages were buried. Greek letters are, at any rate, 
not very well adapted to a vector analysis which aims at 

Maxwell employed German or Gothic type. This was an 
unfortunate choice, being by itself sufficient to prejudice 
readers against vectorial analysis. Perhaps some few readers 
who were educated at a commercial academy where the writing 
of German letters was taught might be able to manage the 
German vectors without much difficulty ; but to others it is a 
work of great pains to form German letters legibly. Nor is 
the reading of the printed letters an easy matter. Some of 
them are so much alike that a close scrutiny with a glass is 
needed to distinguish them, unless one is lynx-eyed. This is a 
fatal objection. But, irrespective of this, the flourishing 
ornamental character of the letters is against legibility. In 
fact, the German type is so thoroughly unpractical that the 
Germans themselves are giving it up in favour of the plain 
Roman characters, which he who runs may read. It is a relic 
of mediaeval monkery, and is quite unsuited to the present 
day. Besides, there can be little doubt that the prevalent 
shortsightedness of the German nation has (in a great measure) 
arisen from the character of the printed and written letters 
employed for so many generations, by inheritance and accu- 
mulation. It became racial ; cultivated in youth, it was inten- 
sified in the adult, and again transmitted to posterity. German 
letters must go. 

Rejecting Germans and Greeks, I formerly used ordinary 
Roman letters to mean the same as Maxwell's corresponding 



Germans. They are plain enough, of course ; but, as before 
mentioned, are open to objection. Finally, I found salvation in 
Clarendons, and introduced the use of the kind of type so- 
called, I believe, for vectors (Phil. Mag., August, 1886), and 
have found it thoroughly suitable. It is always in stock ; it is 
very neat; it is perfectly legible (sometimes alarmingly so), and 
is suitable for use in formulae along with other types, Roman or 
italic, as the case may be, contrasting and also harmonising 
well with them. 

Sometimes block letters have been used ; but it is sufficient 
merely to look at a mixed formula containing them to see that 
they are not quite suitable. I should mention here, however, 
that it is not the mere use of special types that converts scalar 
to vector algebra. For instance, engineers have often to deal 
with vector magnitudes in their calculations, but not (save 
exceptionally) in their vectorial signification. That is, it is 
merely their size that is in question, and when this is the case 
there is no particular reason why a special kind of type should 
be used, whether blocks or another sort. 

In connection with Clarendon type, a remark may be made 
on a subject which is important to the community in general. 
I refer to the retrograde movement in typography which has 
been going on for the last 20 years or so. Many people, who 
possess fairly good and normal eyesight, find a difficulty in 
reading printed books without straining the eyes, and do not 
know the reason. They may think the print is too small, or 
that the light is not good, or that their eyes are not right. But 
the size of type per se has little to do with its legibility ; a far 
more important factor is the style of type, especially as regards 
the fineness of the marks printed. The " old style," revived a 
generation since, and now largely in use, differs from the more 
legible " modern style " (of this page, for instance) in two re- 
spects. It has certain eccentricities of shape, which make it 
somewhat less easy to read ; but, more importantly, the letters 
on the types are cut a good deal finer, which results in a pale 
impression, as if the ink were watery. This is the main cause 
of the strain upon the eyes, and the good light wanted. Even 
very small print is easy to read if it be bold and black, not 
thin and pale. It would be a public benefit if the retrograde 
step were reversed, and the revived old style, which threatens 


to drive out the modern style, discarded. Not that the latter 
is perfect. A still better style would be arrived at by thicken- 
ing the fine lines in it, producing something intermediate 
between it and the Clarendon style (which last, of course, would 
be too much of a good thing). As everyone who has had to 
read MSS. knows, the most legible handwritings irrespective 
of the proper formation of the letters are those in which the 
writing-master's fine upstroke is discarded. Now it is the same 
in print. Thicken the fine lines, and the effect is magical. 

Notation. Tensor and Components of a Vector. Unit 
Vectors of Reference. 

104. The tensor of a vector is its size, or magnitude apart 
from direction. Other important connected quantities are its 
three rectangular scalar components the Cartesian compo- 
nents which are the tensors of the three rectangular vector 
components. It is usual, in Cartesian work, to use three sepa- 
rate letters for these components, as F, G, H ; or u, v, w ; or 
^> fj'&c. One objection to this practice is that when there 
is a large number of vectors the memory- is strongly taxed to 
remember their proper constituents. Another is the prodigal 
waste of useful letters. One alphabet is soon exhausted, and 
others have to be drawn upon. 

In my notation the same letter serves for the vector itself, 
and for its tensor and components. Thus, E denoting any 
vector, its tensor is E, and its components are Ej, E 2 , E 3 . Simi- 
larly, the tensor of a is a, and the components are a lt a^ a y A 
large stock of letters is thus set free for other use. 

But a remark must be made concerning MS. work, as dis- 
tinguished from printed work. In MS. work it is inconvenient 
to be at the trouble of writing two kinds of letters ; ordinary 
letters will suffice for both scalars and vectors. Or the ordi- 
nary letters representing vectors may receive some conventional 
mark to vectorise them. But, presuming ordinary letters are 
written, something is required to distinguish between the vector 
and its tensor. This may be satisfied by calling E the tensor 
of E. Only when an investigation is to be written out for the 
printers is it necessary to bring in special letters, and this is 
most simply done by a conventional mark affixed to every 
e) vector. Compositors are very intelligent, read mathe- 


matics like winking, and carry out all instructions made by the 

If a vector a be multiplied by a scalar x, the result, written 
afa or az, is a vector x times as big as a, and having the same 
direction. Thus, if a 3 be a unit vector parallel to a (or, more 
strictly, parallel to and concurrent with a), of unit length, 
we have a = aa r It is sometimes useful to separately represent 
the direction and length of a vector, and the above is a con- 
venient way of doing it without introducing new letters. This 
applies to any vector. But it need not be an absolute rule. 
For instance, the Cartesian co-ordinates x t y, z of a point may be 
retained. Thus, let r be the vector from the origin to any point, 
and let i, j, k be^ unit vectors from the origin along the three 
rectangular axes. We shall then have 

x = ^i, y = yj, z = zk, . . (1) 

where x is the vector projection of r on the i axis, as we know 
by the elementary geometry of a rectangular parallelepiped or 
brick. Also, the direction-cosines of r are x/r, yjr, z/r, and 

by Euclid I., 47. 

The Addition of Vectors. Circuital Property. 

105. This brings us to vector addition. The vector x 
signifies translation through the distance x in the direction i. 
If, now, after performing this operation, we carry out the 
operation indicated by y, viz., translation through the distance 
y in the direction j ; and, lastly, carry out the operation z, or 
translation through the distance z in the k direction, we shall 
arrive at the end of the vector r. That is, starting from one 
corner of a brick, we may reach the opposite corner by three 
mutually perpendicular journeys along three edges of the brick. 
The final result is the same as if we went straight across from 
corner to corner, that' is, by carrying out the operation indicated 
by the vector r. This equivalence is expressed by 

r = x + y + z, . . , . , . (3) 
Or, by (1), r = XL + yj + zk. 3 , , . (4) 




The meaning of addition of vectors in this example is simply 
the carrying out of the operations implied by the individual 
vectors added, the geometrical vector meaning a displacement 
in space, or translation from one point to another. The order of 
addition is indifferent, since there are six ways of going from 
one corner to the opposite one of a brick along its edges. 

We see that any vector may be expressed as the sum of 
three mutually perpendicular vectors, viz., its vector projections 
on the axes. Furthermore, by the use of a skew parallelepiped 
Instead of a brick we see that any vector may be expressed as 
the sum of three other vectors having any directions we please, 
provided they are independent, or not all in the same plane. 
For in the latter case the parallelepiped degenerates to a plane 

FIG. 2. 

But it is perhaps best to explain vector addition in general 
without any reference to axes. Thus let a and b be vectors to 
be added, a meaning translation from P to Q in the first figure, 
or through an equal distance along any parallel line, as from 
S to R in the second figure, whilst b means translation from Q 
to R in the first figure, or through an equal distance along any 
parallel line, as from P to S in the second figure. In the first 
case, performing the operation a first, and then b, we go from 
P to R irid Q ; in the second case, with b first and then a, we go 
from P to R vid S. The final result in either case is equivalent to 
direct translation from P to R, symbolised by the vector c. 



The above may be extended to any number of vectors. Or we 
may reason thus : Let A be any given vector, translating, say, 


from P to Q. We need not go from P to Q direct, but may 
follow any one of an infinite number of paths, as for example, 



P A 

FIG. 3. 


FIG. 4. 

a + b + c-fdin the figure. The final result of the successive 
translations is always the same, viz., the direct translation A. 

Or A=a+b+c+d (6) 

Thus any vector A may be split up into the sum of any 
number n of vectors, of which n 1 are perfectly arbitrary, for 
instance ,a, b, c in the figure, The remaining one d is, of 
course, not arbitrary. It is the vector required to complete 
the circuit of vectors. 

The vectors need not be in one plane. Nor need the path 
followed consist of finite straight portions. It may be wholly 
or partly curved. The curved portions are then made up of 
infinitesimal vectors. Each curved portion may be replaced 
by the vector joining its terminals. 

Since the original vector and the substituted vectors form a 
circuit, if the positive direction in the circuit be the same for 
all the vectors, we may express vector addition thus: The 
sum of any number of vectors which make a circuit is zero. 
That is, 2a = 0, if a is the type of the vectors summed. For 
a curved circuit we shall have 


where da is the vector element of the circuit. Here s itself 
may be taken to be the vector from any fixed point P to a 
point Q on the circuit, Fig. 4. Then ds is the infinitesimal 


change in s made in an infinitesimal step along the circuit, 
that is, it is the vector element of the circuit itself. 

In a vector equation every term is a vector, of course, how- 
ever the individual terms may be made up, and every vector 
equation expresses the fact symbolised by (7), or by 2a = 
when the vectors are finite. In the latter case, a may be also 
regarded as As, the finite change in the vector s from any fixed 
origin produced by passing from beginning to end of the 
vector a. 

The - sign prefixed to a vector is the same as multiplication 
by - 1, and its effect is simply to reverse the direction of trans- 
lation. Or it may be regarded as reversing the tensor, without 
altering the direction: thus 

- a = - aa x = a x ( - a x ) = ( - a) x aj . . (8). 
Thus, in any vector equation, for example 

a + b + c + d + e + f=0, . . . . (9) 

we may transfer any terms to the other side by prefixing the 
- sign. Thus, 

a + b + c= -d-e-f- -(d + e + f). . . (10) 

In the form (9) we express the fact that translation in a circuit 
is equivalent to no translation. In (10), however, we express 
the equivalence of two translations by different paths from one 
point to another. 

If any trouble be experienced in seeing the necessary truth of 
(9) for a circuit in whatever order the addition be made, the 
matter may be clinched by means of i, j, k, the unit rectangular 
vectors ; as we saw before, the resolution of vectors into 
rectangular component vectors depends only upon the properties 
of the right angled triangle. Thus split up the vectors a, b, c 
into components, we have 

k6 3 , ..>.';. (11) 
c = icj + jc 2 + kc 3 , ) . 

Now add up. All vectors parallel to i add in scalar fashion, 
for there is no change of direction. Similarly for j and k ; so 
we have 


Now the scalar additions may be done in any order we 
please. It follows that the same is true in vector addition. 
The addition and subtraction of vectors and transfer from one 
side of an equation to the other are thus done identically as in 
the algebra of scalars. Addition has, it is true, not the eame 
meaning, but the vector meaning is not inconsistent with the 
scalar meaning, and, in fact, includes the latter as a particular 
case. There is never any conflict between + put between 
vectors and + put between scalars. 

Application to Physical Vectors. Futility of Popular Demon- 
strations. Barbarity of Euclid. 

106. In the above we have referred entirely to the geo- 
metrical vector. But a very important step further can be 
made referring to physical vectors, a step which immediately 
does away with piles of ingeniously constructed and brain- 
wasting " demonstrations," especially compiled for the tortur- 
ing of students and discouragement of learning. If a quantity 
be recognised to be a directed magnitude, it is, in its mathe- 
matical aspect, a vector, and is therefore subject to the same 
laws as the vector, or geometrical vector. So, just as we have 
the triangle, or parallelogram, or polygon of geometrical vectors, 
we must have the same property exemplified in the addition of 
all vectors, as velocity, acceleration, force, &c. This identity of 
treatment applies not only to the addition property, but to the 
multiplication and other properties to be later considered. 
We may symbolise a physical vector by a straight line of given 
length and direction. 

It used to be thought necessary to give demonstrations of 
the parallelogram of forces, perhaps even before the student 
knew what force meant. I have some dim recollection of days 
spent in trying to make out Duchayla's proof, which was 
certainly elaborate and painstaking, though benumbing. Max- 
well, in his treatise, elaborated a demonstration that electric 
currents compounded according to the vector law. But surely 
there is something of the vicious circle in such demonstrations. 
Is it not sufficient to recognise that a quantity is a vector, to 
know that it follows the laws of the geometrical vector, the 
addition property of which does not want demonstrating, but 
only needs pointing out, as in the polygon of vectors ? 



Although this is not the place for exercises and examples, yet 
it is worth while to point out that by means of the addition 
property of vectors a good deal of geometry can be simply 
done better than by Euclid, a considerable part of whose 12 
books consists of examples of how not to do it (especially 
Book V.). There is a Society for the Improvement of Geometrical 
Teaching. I have no knowledge of its work ; but as to the 
need of improvement there can be no question whilst the reign 
of Euclid continues. My own idea of a useful course is to begin 
with arithmetic, and then, not Euclid, but algebra. Next, not 
Euclid, but practical geometry, solid as well as plane ; not 
demonstrations, but to make acquaintance. Then, not Euclid, 
but elementary vectors, conjoined with algebra, and applied to 
geometry. Addition first; then the scalar product. This 
covers a large ground. When more advanced, bring in the 
vector product. Elementary calculus should go on simultane- 
ously, and come into the vector algebraic geometry after a bit. 
Euclid might be an extra course for learned men, like Homer. 
But Euclid for children is barbarous.* 

The Scalar Product of Two Vectors. Notation and 


107. Coming next to the products of vectors, it is to be 
noted at the beginning that the ordinary idea of a product in 

* From The Electrician, December 4, 1891, p. 106, I learn that the 
correct title of the society above alluded to is the Association for the 
Improvement of Geometrical Teaching. " It was founded, we believe, 
about ten years ago by a few teachers, who realised that Euclid for 
children is, as Mr. Heaviside puts it, simply barbarous. Three pamphlets 
have been published by Messrs. Macmillan and Co. a syllabus of plane 
geometry, corresponding to Euclid, Books I. to VI. ; another of modern 
plane geometry ; and another of linear dynamics. Messrs. Swan, Sonnen- 
schein and Co. have published an elementary geometrical conies, and there 
the labours of the Association appear to have stopped. The Association 
has had to struggle against the stubborn conventionalism of the modern 
schoolmaster, who pleads that he cannot make any change, because of the 
Universities. After a considerable fight, Mr. Hamblin Smith's common- 
sense proofs of Euclid's problems were accepted by Cambridge examiners. 
Small as the visible results of the Association have been, there is a distinct 
change of feeling taking place with regard to geometry, both as an educa- 
tional subject and as an implement of scientific work. At present geometry 
is taught as badly as Greek, even in the best public schools ; and the 
educational value of Greek is in many respects higher than that of Euclid." 


arithmetic and in scalar algebra does not apply to vectors, 
because they are not scalars. We cannot, therefore, say before- 
hand what the product of two vectors ought to be, or deter- 
mine this question by any prior reasoning of a legitimate nature. 
We must examine in what way vectors enter into combination 
as products, and then introduce definitions and conventions of 
notation to give expression to the facts in the simplest and 
most convenient way. After this, the work is deductive. 

By this examination we are led to recognise two distinct 
kinds of products of a pair of vectors, the scalar product and 
the vector product. It is with the former of these that we are 
now immediately concerned. 

We define the scalar product of a pair of vectors A and B 
whose tensors are A and B, and whose included angle is 6, to 
be the scalar AB cos 0, and we denote it by AB. Thus 

AB = ABcos0 (12) 

defines the scalar product and its notation. 

To see its full significance and how it works out, let us first 
apply (12) to the unit reference vectors, i, j, k, which are oeper- 
pendicular, by taking A and B to be one or other of them in 
turn. First multiplying each of i, j, k by itself, we get 

12 = 1, J2 = l, k2=l, . . . . (13) 

with the convention borrowed from scalar algebra that ii is 
equivalently denoted by i 2 . This may be called the square 
of i. In each of the three cases (13) the vectors multiplied 
together are of unit length, and are parallel and concurrent, so 
that the scalar product, according to (12), is unity. 

Similarly, the square of any unit vector is unity. And since 
A = AA 15 the square of any vector is the same as the square of 
its tensor ; or 

A 2 = A 2 (14) 

Also, the scalar product of any two parallel vectors is the 
product of their tensors, for AB = AB A 1 B 1 , and A 1 B 1 is now 

Next make scalar products of i, j, k in pairs. We get 

ij = 0, jk = 0, ti = (15) 


That is, the scalar product of two perpendicular unit vectors is 
zero. The same is, of course, true of any two perpendicular 
vectors. Thus the equation AB = means that A is perpen- 
dicular to B ; unless, indeed, one or other of them is zero. 
We have also ^ 

AA-cosfl, (16) 

by dividing (12) by AB ; or, the scalar product of any two 
vectors is the cosine of the included angle. This is reckom 
positively from concurrent coincidence, so that as ge2s from 
to 2 TT, AjBi goes from 1 through to 1, and then through 
again to + 1. 

There is, strictly, no occasion to introduce trigonometry. Or 
we might make the trigonometry be a simultaneously developed 
subject. It is, in fact, a branch of vectorial algebra, being 
scalar developments of parts thereof. We may employ the idea 
of perpendicular projection simply. rrThus, we may say that the 
scalar product of a pair of unit vectors is the length of the 
projection of either upon the other ; and that the scalar 
product of any vector A and a unit vector i is the projection of 
A upon the axis of i ; and, comprehensively, that the scalar 
product AB of any two vectors A and B is the product of the 
tensor of either into the projection (perpendicularly) upon it of 
the other. It is the effective product, so to speak H In physical 
mathematics scalar products frequently have reference to 
energy, or activity, or connected quantities. Thus, if F be a 
force and v the velocity of its point of application, their scalar 
product Fv is the activity, of the force ; it is the product of the 
speed and the effective force. When the force and the velocity 
are perpendicular, the activity is nil, although the velocity may 
be changing a fact which familiarity does not render less 
striking. When F and v are parallel (whether concurrent or 
not), Fv becomes the same as Fi;, in the common meaning of a 
product. A notation^that harmonises in this way is obviously 
a convenient one. 

Notice, also, that AB = BA, as in scalar algebra. 

A frequently occurring operation is the surface integral of 
the normal component of a vector ; for example, to express the 
amount of induction through a surface. Here the idea of a 
unit normal vector N is useful. The normal component of B 


is then NB, and: the integral is 2 NB, the summation extending 
over the surface. 

Similarly, to express the line-integral of the effective 
component of a vector along a line, we may let T be the unit 
element of curve ; that is, the unit tangent ; then TE is the 
tangential component of the vector E, and 2TE is the integral ; 
for example, the voltage between two points along the path to 
which the summation refers, if E be the electric force. 

Since ab is a scalar, it behaves as a scalar, when considered 
as a whole. Thus, when multiplied by a scalar, x, the result, 
#ab or aba:, is scalar, being simply x times ab. When multi- 
plied by a vector the result is a vector ; thus, c.ab or ab.c 
means ab times the vector c . The dot here acts rather as a 
separator than as a sign of multiplication. Thus, to illustrate, 
ca.b means ca times the vector b ; and, similarly, a.bc is be 
times the vector a. But, instead of the dot, we may use 
brackets to indicate the same thing, thus c.ab may be written 
c(ab). This is, perhaps, preferable in initiatory work, but I 
think the dot plan is more generally useful. 

We have an example of this combination of three vectors in 
the stress formulae. Thus, the electric stress vector on the 
plane whose unit normal is N, is expressed by 

E.DN - N.JED, 

(equation (31), 72) ; that is, the sum of two vectors, parallel 
to E the electric force and to N respectively, whose tensors are 
EDN and -JED respectively, where D is the displacement. 
The interpretation as a tension along E combined with an equal 
lateral pressure, obtained by taking N parallel to and then per- 
pendicular to E or D, has been already discussed. 

Fundamental Property of Scalar Products, and Examples. 

108. As all the preceding is involved in the definition of a 
scalar product, and is obvious enough, it may be regarded 
merely as illustrative. The reader might, in fact, say that ha 
knew it all before in one form or another, trigonometrical or 
geometrical, and that he did not see any particular advantage 
in the way of stating it in the notation of scalar products. 
But we now come to a very striking and remarkable property 


of scalar products, which will go far to justify them as working 

We know that in scalar algebra, when we have a product 
xy t we may express x by the sum of any number of other 
quantities, and similarly as regards y, and then obtain the 
complete product xy by adding together all the component 
products obtained by multiplying every element of x into every 
element of y ; and that this process may be carried out in any 

Now, in vector algebra, we know already that there is a 
partial similarity, viz., that we can decompose a vector A into 
the sum of any number of others, and similarly as regards B. 
The question now is whether the scalar product AB is the sum 
of all the scalar products made up out of the components of A 
paired with the components of B, taken in any order. For 
example, if 

and B = c + d, 

is AB = a + bc + d ) 

J * 

The answer is Yes, and the process of manipulation is the 
same as in scalar algebra, that is, as if all the vectors were 
scalars. Moreover, this property admits of demonstration in 
a sufficiently simple manner as to enable one to see its truth. 

Start with the vector A, and first split it up into 

A = a + b + c, ..... (18) 

two or three vector components being sufficient for illustration. 
Now project the vector A perpendicularly upon any axis, say 
that of i. According to our definition of a scalar product, the 
projection is Ai. ' Now it requires no formal demonstration, 
but becomes evident as soon as the meaning of the proposition 
is correctly conceived, that the projection of A upon any axis 
is the same as the sum of the projections of its component 
vectors, a, b, c, on that axis. That is, the latter projections 
are either all positive or all negative, and fit together to make 
up the projection of A ; or else some may be negative and 
others positive, when there is overlapping and cancelling, but 


till with the same result algebraically. This is expressed by 
Ai = (a + b + c)i = ai + bi + ci, . . . (19) 

; got by multiplying (18) by i. 

If we now multiply (19) by any scalar B, so that Bi=B, 
which is any vector, since i may have any direction, we obtain 

AB = (a + b + c)B = aB + bB + cB, . . (20) 

the same as if we multiply (18) by B direct. 

Similarly, we may split up B into the sum of any number of 
other vectors, say, 


If we substitute this in (20) we have 

Now make use of the same reasoning which established (19) 
and (20), applied to the bracketed vectors, and we establish 
the property fully, with the result 

= ad + ae + af + bd + be + bf + cd + ce + cf ; 

the expansion being done formally as in scalar algebra in every 
respect ; since the various terms may be written in any order, 
and each may be reversed, thus, ad = da. 

I have already remarked that a good deal of geometry may 
be done by the addition property alone. The range of applica- 
tion is greatly extended by the use of the scalar product and the 
fundamental property (17). 

To obtain the Cartesian form of AB, put the vectors in terms 
of i, j, k; thus 

AB = (A 1 i + A 2 j+A 3 k) (B^ + BJ + Bak). . (22) 

Now effect the multiplications, remembering (13) and (15). 
The result is 

AB = A 1 B 1 + A 2 B 2 + A 3 B 3 . . . . (23) 

For example, the activity of a force is the sum of the 
activities of its component forces in any three coperpendicular 
directions. Or, if A and B are unit vectors, we express the 
cosine of the angle between them in terms of the products of 


the direction cosines of the vectors, each product referring to 
one axis. In (23) we may, if we please, vectorise the six scalars 
on the right side. 

Since the square of a vector is the square of its tensor, we 
may express the tensor at once in terms of the tensors and the 
cosines of the angles between a series of vectors into which -the 
original vector is resolved. Thus, for two, 


(A-B) 2 = A 2 -2AB + B 2 ; . . . (25) 
and similarly for three, 


Here (24) and (25) apply to a parallelogram, and (26) to a 
parallelepiped. In (24), (25), if A and B be the vector sides of 
a parallelogram, then (A + B) and (A-B) are the vector 
diagonals ; so (24) gives the length of one diagonal and (25} 
that of the other. 

Adding (24) and (25) we obtain 

-B) 2 = 2(A 2 + B 2 ), . . (27) 

expressing that the sum of the squares of the diagonals equals 
the sum of the squares of the four sides. Similarly, by sub- 

(A + B) 2 -(A-B) 2 = 4AB, . . . (28) 

expressing the difference of the squares of the diagonals as four 
times the scalar product of two vector sides. 

Equation (27) also shows that the sum of the squares of the 
distances of the ends of any diameter of a sphere from a fixed 
point is constant. For if A is the vector from the fixed point 
to the sphere's centre, and B the vector from the centre to one 
end of the diameter, then A + B and A-B are the vectors- 
from the fixed point to the ends of the diameter. Whence, by 
(27), the proposition. 

There is an application of this in the kinetic theory of gases* 
For when two elastic spheres collide they keep the sum of their 
kinetic energies constant. If, then, their velocities before 
collision be A + B and A-B, their velocities after collision are 
indicated by vectors from the origin (in the velocity diagram) 


to the ends of some other diameter of the sphere described 
upon the line joining their original positions (in the velocity 
diagram) as diameter ; which is actually the new diameter 
depending upon the circumstances of impact. 

Reciprocal of a Vector. 

109. It is occasionally useful to employ the reciprocal of a 
vector in elementary vector algebra. We define the reciprocal 
of a vector a to be a vector having the same direction as a, 
and whose tensor is the reciprocal of that of a. We may denote 
the reciprocal of a by a -1 or I/a. Thus as a = aa p we have 


Any unit vector is, therefore, its own reciprocal. 

The reciprocal of a vector, being a vector, makes scalar pro- 
ducts with other vectors. Thus ab- 1 or a/b means the scalar 
product of a and b- 1 , and we therefore have 

i-^-S-H-S- ' (30) 

The tensor of a/b is (a/b) cos 6, where 6 is the angle between 
a and b, or between their reciprocals, 'or between either and the 
reciprocal of the other. 

So a/a or aa' 1 or a' 1 a equals unity. 

In using reciprocals the defined meaning should be at- 
tended to, especially when put in the fractional form. Thus we 
easily see that aV 1 = ab/6 2 , because the tensor of b is b 2 times 
the tensor of b' 1 . But we cannot equivalently write a 2 /ab, 
because this is (a/6)/cos 0, which is quite a different thing. 

Notice that a" 1 b- 1 . is not the same as (ab)- 1 . The first is 
a- 1 6' 1 cos 6, whilst the latter is or 1 ft^/cos 0. 

Expression of any Vector as the Sum of Three Independent 


110. We know that in the equation 

i yd -i "^"j 

where r is the vector distance of a point from the origin, the 
scalars are the lengths of the projections of r upon the axes. 


How should we, however, find them in terms of r algebraically ? 
To find x we must operate on the equation in such a manner as 
to cause the j and k terms to disappear. Now this we can do 
by multiplying by i. For i is perpendicular to j and k, so that 
multiplication by i gives 

Similarly, rj = y and rk = z. 

From this obvious case we can conclude what to do when the 
reference vectors are not perpendicular ; for instance, in 

...... (31) 

where a, b, c are any independent vectors. To find / we must 
multiply by a vector perpendicular to b and c, say 1, so that 
Ib = 0, and Ic = 0. Then 

rl=/al, therefore /=rl/al 
Similarly to isolate g and A, so that we get 

'-i'OO ..... (32> 

where 1, m, n are vectors normal to the three planes of b,c, c,a 
and a,b. This exhibits explicitly the expansion of any vector 
in terms of any three independent vectors a, b, c, as the three 
edges of a skew parallelepiped. This ease, of course, reduces 
to the preceding Cartesian case by taking a, b, c to be i, j, k, 
when 1, m, n will also be i, j, k, or any scalar multiples of the 

Observe the peculiarity that auxiliary vectors are used, each 
of which is perpendicular to two others, that is, to the plane 
containing them. These auxiliary vectors bring us to the 
study of the vector product. 

The Vector Product of Two Vectors. Illustrations. 

111. The auxiliary vectors just employed in the expansion of 
a vector into the sum of three vectors having any independent 
directions, are examples of vector products. Two vectors being 
given, their vector product is perpendicular to both of them. 
Of course, disregarding magnitude, there is but one such 



vector, viz., the normal to the plane containing the given 
vectors. Nor is the tensor of any consequence in the example 
in question, for it will be observed that one of the vectors, 
1, m, n, appears both in the numerator and in the denomi- 
nator of one of the three fractions in equation (32), so that 
the values of the fractions are independent of the tensors of 
the auxiliary vectors. 

But the vector product of two vectors has a strictly-fixed 
tensor, depending upon those of the component vectors and 
their inclination. There is particular advantage in taking the 
tensor of the vector product of a and b to be ab sin 0. Thus 
we define the vector product of two vectors a and b whose 
tensors are a and &, and whose included angle is 0, to be a 


FIG. 5. 

FIG. 6. 

third vector c whose tensor c equals ab sin 0, and whose direc- 
tion is perpendicular to the plane of a and b ; the positive 
direction of c being such that positive or right-handed rotation 
about c carries the vector a to b. This vector product is 
denoted by 

c = Vab, ...... (33) 

and its tensor may be denoted by V ab, so that we have 


Similarly, V x ab may be used to denote the unit vector 
parallel to Vab. 

The only troublesome part is to correctly fix which way along 
the perpendicular to the plane of a and b is to be considered 
positive. Examples will serve to clinch the matter. Thus, let 


a be towards the north, and b towards the east on the earth's 
surface ; then c is straight downwards. Again, in Fig. 5, the 
direction of c is downwards through the paper, and its tensor 
is the area of the parallelogram upon a and b. 

Let b be fixed, whilst a moves round so as to vary the angle 
Q. Starting from coincidence, with 6 = 0, the tensor of c is zero. 
It reaches a maximum (downwards) when is a right angle, 
and falls to zero again when reaches two quadrants, and a is 
in the same line with b, though discurrent. After this, in the 
next two quadrants, the same numerical changes are gone 
through ; but now, the sine being negative, c must be upwards 
from the paper. 

The unit reference vectors i, j, k are so arranged that when, 
in (33), a is i and b is j, then c is k ; noting in Fig. 6, that k 
is supposed to go downwards, or away from the reader. Thus, 
in accordance with our definition we have 

Vij = k, Vjk = i, Vki=j, . . (35) 

because the mutual angles are quadrants and the tensors unity. 
Observe the preservation of the cyclical order i, j, k in (35). 
Also, we have 

Vii = 0, Vjj = 0, Vkk = 0, . . (36) 

because the vectors paired are coincident. 

By the definition, the reversal of the order of the letters in 
a vector product negatives it. Thus . 

Vab= -Vba. ..... (37) 

Since the tensor of Vab is ab sin and the scalar product 
.ab is ab cos 0, we have 


Combinations of Three Vectors. The Parallelepipedal 

112. A vector product, being a vector, of course combines with 
other vectors to make scalar and vector products again. Thus 
cVab, where c is any new vector, means the scalar product of c 
and Vab; and VcVab means the vector product of c and Vab. 
These are both important combinations, which occur frequently, 
and their interpretations and expansions will be given presently. 


As illustrative of notation, it may be mentioned that cV ab, 
where, as before explained, V ab is the tensor, is obviously a 
vector parallel to c, but V ab times as long. On the other 
hand, cVjab, where V x ab is the unit vector, is the same as 
cVab/V ab; since by dividing by its tensor we unitise a 
vector. Similarly as regards VcVjab. We may also have 
V^Vab and V^V^b and V cVab, and various other modifica- 
tions, whose meanings follow from the definition of a vector 
product and its notation. 

We do not often go further in practical vector algebra than 
combinations in threes ; for instance, on to dVaVbc, the scalar 
product of d and the previously explained VaVbc. 

The scalar product cVab has an important geometrical illus- 
tration. Its value is given by 

cVab = V abxccos<, .... (39) 

where < is the angle between c and Vab. This is by the defi- 
nition of the scalar product, and of the tensor of a vector. 

Now refer to Fig. 5 again. We know that V ab is the area 
of the parallelogram. We also know that the volume of a 
parallelepiped is the product of its base and altitude. Construct, 
then, a parallelepiped whose three edges meeting at a corner 
are a, b, and c. The area of one of its bases is V ab, and the 
corresponding altitude is c cos <. Therefore, by (39), cVab is 
the volume of the parallelepiped. 

But there are two other bases and two other altitudes to 
correspond, so there are two other ways of expressing the 
'volume, giving the equalities 

.... (40) 

in which observe the preservation of cyclical order, done to 
keep the sign the same throughout, as will be verified a little 

Semi-Cartesian Expansion of a Vector Product, and Proof 
of the Fundamental Distributive Principle. 

113. The semi-Cartesian expansion of c = Vab is 

C = i(a 2 6 3 -a 3 6 2 )+j(a 3 6 1 -a 1 6 3 )+k(a 1 6 2 -a 2 6 l ), . (41) 
in terms of i, j, k and the scalar components of a and b. 


To prove this, multiply (41) by a and b in turns to form the 
scalar products ac and be. Do this by the rule embodied in 
equation (23). We get 

ac = aj (a 2 6 3 
be = 6j (a 2 6 ;i 

But, by cancelling, all the terms on the right disappear, 
That is, 

ac = 0, be = 0. 

From these we know that c is perpendicular to a and to b. 
It is, therefore, Vab itself, or a multiple of the same. To find 
its tensor, square (41). We get 

c 2 = (aj), - 0362)8 + (a j h _ ai j 3 )2 + (ai&2 _ a2 j i)a . (42) 
But this may, by common algebra, be transformed to 

c 2 = fa* + a* + a 3 2 ) (V + Z> 2 2 + & 3 2 ) - (a^ + a 2 & 2 + 3 5 3 ) 2 . (43) 
That is, c 2 = a 2 b 2 - (ab) 2 , ..... (44) 

or, c 2 = a 2 6 2 - a 2 6 2 cos 2 6 = (ab sin 0) 2 . 

The tensor c is, therefore, that of Vab itself, or else its 
negative. Equation (44) is the same as (38), in a slightly 
different form. Equation (41) is proved, except that it remains 
to be seen whether the right member represents Vab or its 

To do this, we may remark that the angle between two 
planes is the same as the angle between the normals to the 
planes, if the normals coincide when the planes do, so that the 
three quantities in ( )'s in (41), which are the scalar pro- 
jections of c on the axes, are also the areas of the projections 
of the area V ab on the planes perpendicular to them. We 
have therefore only to verify that the projection of any plane 
on the plane normal to one of the axes is given correctly in 
sign by (41). Take a rectangle, parallel to the j, k plane, 
sides parallel to j and k, that is, 2 j and 6 3 k ; so that 

Here 2 6 3 is the projection of the rectangle on a parallel plane, 
and is correctly positive. 


Now go back to (41). We have 

Vab = i(a 2 6 3 -a 3 6 2 )-i- (45) 

Similarly Vac = i (a 2 c 3 - a 3 c 2 ) + (46) 

Add these together. We get 

Vab + Vac = i[a 2 (6 3 + c 3 )-a 3 (6 2 + c 2 )] + (47) 

But the right member, by definition of a vector product, or by 
(41 ), means Va (b + c). That is, 

Va(b + c) = Vab + Vac. .... (48) 

Similarly Vd (b + c) = Vdb + Vdc (49) 

So, by adding these again, 

V (a + d) (b + c) = Vab + Vac + Vdb + Vdc, . (50) 

a very remarkable and important formula. It shows that the 
vector product of two vectors, A and B, equals the sum of all 
the vector products which can be made up out of the com- 
ponent vectors [(a + d) and (b + c) in (50)] into which we may 
divide A and B, provided we keep the components of A always 
before those of B. The necessity of this proviso of course 
follows from the reversal of Vab with the order of a and b. 
Although (50) only shows this for two components to each 
primary vector, yet the process by which it was obtained evi- 
dently applies to any number of components. 

Subject to the limitation named, the formula (50) is pre- 
cisely similar to (17) 108, relating to the scalar products. 
Now the truth of (17) could be seen without much trouble. 
A similar proof of (50), on the other hand, would not be at all 
easy to follow, owing to the many changes of direction in- 
volved in the vector products. This is why I have done it 
through i, j, k, which are auxiliaries of the greatest value. 
When in doubt and difficulty, fly to i, j, k. 

On the other hand, although the establishment of (50) by 
geometry without algebra is difficult, and there is preliminary 
trouble in fixing the direction of a vector product, yet we see 
from (50) that vector products are nearly as easily to be 
manipulated algebraically as scalar products. 


Examples relating to Vector Products. 

114. By means of the formula (41) we can at once obtain 
the cartesian expansion of the parallelepipedal cVab. Use 
equation (23), applied to (41), remembering that c is now any 
vector, and we get 

cVab = ^(0363 - a 3 6 2 ) + c^a^ - a^) + c^a^ - aj)^ . (51) 

By rearrangement of terms on the right side of this, putting 
the a's outside the brackets, we obtain aVbc, and putting the 
b s outside, we obtain bVca, and thus verify (40) again. 

The vector VcVab is, perhaps, most simply expanded through 
i, j, k. First write d for Vab, then by (41) we have 

Vcd = icd 

Next put for d v d 2 , d 3 their values in terms of a's and b's, 
given in (41), and we have 

Next, add and subtract a^c^ and we get 

VcVab = ifc^bc - ^ac] + . . . . 
Lastly, reconvert fully to vectors, and we have 

VcVab = ..... (52) 

2 *^ "i. ^ % I 2* 

This important formula should be remembered. That VcVab 
is in the plane of a and b is evident beforehand, because Vab 
is perpendicular to this plane, and multiplying by Vc sends it 
back into the plane. It is, therefore, expressible in terms of a 
and b, as in (52), which shows the magnitude of the com- 

In some more complex formulae it is sufficient to remember 
the principle upon which they are founded, as by its aid they 
can be recovered at any time. Thus in the case of 

r=/a+#b + Ac, ..... (53) 

already treated, equation (31), if we use now Vbc for 1, which 
was, before, any multiple of it, and similarly for m and n, we 

have r 


There is no occasion to put a formula like this in the 
memory, because it is so simply obtained at any time from 
(53) by multiplying by the auxiliary vectors so as to isolate /, 
<7, h in turn and give their values. 

Notice that the three denominators in (54) are equal. Also 
that by exchanging a and Vbc, b and Vca, c and Vab, we 

r- " Vbc+ * Vca+--Vab, . (55) 
aVbc bVca cVab 

and this is also true, as we may at once prove by multiplying 
it in turns by a, b, and c, each of which operations nullifies two 
terms on the right. (55) is the expression of r in terms of 
three vectors, which are normal to the three planes of any 
three independent vectors, a, b, c, taken in pairs. Here, again, 
such a formula can be immediately recovered if wanted by 
attending to the principle concerned. 

To obtain the cartesian expansion of any formula containing 
vectors is usually a quite mechanical operation. The cartesian, 
or semi-cartesian, representatives of a few fundamental functions 
being remembered, their substitution in the vector formula is all 
that is required. If the formula be scalar (although involving 
as many vectors as we please) the result is a single scalar for- 
mula in cartesians. But if it be a vector formula, it reduces 
to a semi-cartesian vector formula involving i, j, k, giving 
three scalar equations, one for each component. 

The converse process, to put a scalar cartesian investigation 
into vectorial form, is less easy, though dependent upon the 
same principles. Here the three component equations have to 
be reduced to one vector equation. It is very good practice to 
take a symmetrically written-out cartesian investigation and 
go through it, boiling it down to a vector investigation, using 
the unit reference vectors i, j, k, whenever found to be con- 

The Differentiation of Scalars and Vectors. 

115. In the preceding account of vector addition, and of the 

scalar product and the vector product, the reader has nearly all 

that is needed for general purposes in geometry and in the usual 

physical mathematics involving vectors, so far as the algebra 

M 2 


itself is concerned. For the addition of differentiations does 
not usually introduce anything new into the algebra. Thus, in 
the analysis of varying vectors, we have the same vector 
algebra with new vectors introduced, these being derived from 
others by the process of differentiation. 

The ideas concerned in the differentiation of a vector with 
respect to a scalar are essentially the same as in the differentia- 
tion of a scalar. Thus, u being a scalar function of x, i.e., a 
quantity whose value depends on that of x, we know that if 
AM is the increment in u corresponding to the increment A# 
in x, then the ratio Aw/A# usually tends to a definite limiting 
value when A# is infinitely reduced, which limit, denoted by 
du/dxy is called the differential coefficient of u with respect to x. 
This I should be strongly tempted to call the " different iant " 
of u to x, were I not informed on the highest authority that 
the expression " differentiant " for "differential coefficient" is 
objectionable. But the differentiating operator djdx which 
acts on the operand u may be termed the " differentiator," as 
has perhaps been done by Sylvester and others. This way of 
regarding a differential coefficient, splitting du/dx into (d/dx) 
and u, sometimes leads to great saving of labour, though we are 
not concerned with it immediately. 

The differential coefficient is thus strictly the rate of increase 
of the function with the variable, or the increase of the func- 
tion per unit increase of the variable, .on the tacit assumption 
that the rate of increase of the function keeps the same value 
throughout the whole unit increment in the variable as it has 
for the particular value of the variable concerned. This plan, 
referring to unit increment, is often very useful. The reservation 
involved, though it must be understood, need not be mentioned. 

There is also the method of differentials, which is of some 
importance in vector analysis, as it can be employed when 
differential coefficients do not exist. Thus, u being a function 
of x, whose differential is dx, the corresponding differential of 

du=f(x + dx)-f(x); . .'.. . (56) 
which, by expanding/ (x + dx), reduces to 

dudx, (57) 


provided we neglect the squares and higher powers of dx, that 
is, regard dx as infinitesimal, and therefore du also. And if 
there be two variables, as in u=f(x, y), then we have 

f(x,y), . . (58) 

leading to du = dx + dy, ..... (59) 

dx dy 

expressing the differential of u in terms of those of x and y. 

Now let the operand be a vector function of x t say, E. For 
instance, x may mean distance measured along a straight line 
or axis in an electrostatic field, where E, the electric force, will 
usually change as we pass along the line. It may change con- 
tinuously in direction as well as in magnitude ; in any case, the 
change itself in E between two points on the line is a vector, 
being the vector AE which must be added to the E at the first 
point to produce that at the second point. Dividing by Ax, the 
increment in x, we get the vector AE/A# ; and this, when the 
increments are taken smaller and smaller, approximates towards 
a definite limiting vector denoted by dE/dx, which is, by the 
manner of its construction, the rate of increase of E with x, or 
the increase in E per unit increase in x. 

We have also, in differentials, 

-f(x), .... (60) 

leading to dE=^dx, ...... (61) 


understanding that the differentials dx and dH are infinitesimal. 
Similarly, the rate of increase of the vector d'E/dx with x is 
the second differential coefficient d 2 E/dx 2 ; and so on. 

Semi-Cartesian Differentiation. Examples of Differentiating 
Functions of Vectors. 

116. In semi-Cartesian form we have 

E = iE 1 +jE 2 + kE 3 ..... (62) 

Here the scalars E I} E 2 , E 3 are functions of x, whilst the 
reference vectors i, j, k are constants ; so we have 

^? = i^l+j^2 + k ^3 j . (63) 

dx dx dx dx 

showing the components of the vector dEjdx. 


Snarly, i +J . . . (64) 

dx* ax 2 ax 2 ax* 

shows the components of the second differential coefficient, 
obtained by differentiating (63). 

Now, this semi-Cartesian process is quite general. Any vector 
function may be expressed in the form (62), and when the right 
member is differentiated, the differentiations are performed 
upon scalar functions. The same remark applies to (63) and 
(64), &c. ; so we see that the rules for differentiating vectors, 
and functions of vectors, with respect to scalar variables, 
are the same as those for differentiating similar functions of 
scalars, so far as the independent action of a differentiator on 
the separate members of a product goes. 

Thus, in differentiating a scalar product, AB, with respect to 
a variable scalar, t, we have, by (23), 

= (A! B! + A 2 B 2 + A 3 B 3 ) + (A l B x + A 2 B 2 + A 3 B 3 ) ; 
or, re-transforming to vectors, 

just as if A and B were scalars. Similarly, we have 

.... (66) 

as we may see immediately by differentiating the semi-Cartesian 
expansion of VAB, equation (41). We have also 

^AVBC = AVBC + AVBC + AVBC, . . . (67) 

and A VAVBC = VAVBC + VAVBC + VAVBC. . . (68) 

But, although we proceed formally, as in the ordinary 
differentiation of scalars, as regards the properties peculiar to 
differentiation, yet we must always remember those which are 
peculiar to the vector algebra. For example, we must not 
only on the left sides, but also on the right sides of (67) and 


(68), remember the reversal of the sign of a vector product with 
the order of the letters. 

Independently of i, j, k, we may proceed thus, using differen- 
tials : 

d (AB) = (A + cf A) (B + dB) - AB = AdB + Bo?A, . (69) 

omitting the infinitesimal dAdE of the second order. Dividing 
by dt, and proceeding to the limit, we obtain (65). 
Similarly we have 

d (VAB) = V(A + dA.) (B + dB) - VAB = V(</A)B + VAdB, . (70) 

omitting the second differential WAc?B. So, dividing by dt, 
we obtain (66). 

In getting (69) we have, of course, used the distributive 
law of scalar products, equation (17) ; and, in getting (70), the 
similar law of vector products, equation (50). 

Motion along a Curve in Space. Tangency and Curvature ; 
Velocity and Acceleration. 

117. Some spacial and motional examples maybe here use- 
fully inserted to illustrate the differentiation of vectors. Let 

FIG. 7. 

there be any curve in space, and let s be length measured along 
it from some point on the curve. Also let r be the vector from 
any fixed origin to a point P on the curve. We have then 
r =/(s). With the form of the function we have no concern. 

Now, consider the meaning of the first differential coefficient 
dijds. First, if the increment As is finite, say the arc PQ, the 


corresponding increment in Ar is the vector chord PQ, the new 
vector of the curve being OQ, that is r + Ar, instead of OP orr. 
We see that Ar/As is a vector parallel to the chord, whose tensor 
is the ratio of the chord to the arc. Now reduce the incre- 
ments to nothing. In the limit the direction of dr/ds is that 
of the tangent to the curve at P, and its tensor is unity, since 
the chord and the arc tend to equality. That is, 

J-I-irff + j^ + k*, . . . .(71) 
as as as as 

where T is the unit vector tangent, and the semi-Cartesian 
expression (got by expanding r) is also given. In the figure the 
size of T is quite arbitrary, since the unit of length is arbitrary. 

Next, differentiate again with respect to s to obtain the 
second differential coefficient d 2 T/ds 2 or dT/ds. It is the change 
in the unit tangent per unit step along the curve. Now, as the 
unit tangent cannot change its length, it can only change its 
direction ; and, moreover, this change is a vector perpendicular 
to the tangent. It and the two tangents, initial and final, are 
in some plane, usually termed the osculating plane. It is the 
plane of the curve for the time being, unless, it be a plane 
curve, when it is the plane of the whole curve. Thus dT/ds, 
being at right angles to T, points from the curve towards the 
centre of curvature for the time being. Moreover, its tensor 
measures the curvature. For the usual measure is d6/ds t 
where dO is the angle between the tangents at the ends of ds, 
and it may be readily seen that this is the tensor of dT/ds. 

Now the reciprocal of the curvature is the radius of curva- 
ture. If then R be the vector from the curve to the centre of 
curvature C (Fig. 7), we have 

& -5 3 

and the vector from the origin to the centre of curvature is 
r + R. 

Now, referring to the same figure, let a point move along the 
curved path. Consider its velocity and acceleration, taking t 
the time for independent variable. We have 

* = **-t,T = v,. (73) 

dt ds dt l ; 


by using (71), and denoting the velocity by v. Its tensor is 
v or ds/dt. The interpretation of (73) is sufficiently obvious, 
since it says that the motion is (momentarily) along the 
tangent, at speed v. 

Differentiate again to t. We get 

. (74) 

dt 2 dt dt 

That is, the vector rate of acceleration is exhibited as the sum 
of two vectors, the first being the tangential component, whose 
tensor equals the rate of acceleration of speed, whilst the second 
we may expand thus, by introducing the intermediate vari- 
able s : 

&-,*_* ..... (75) 
dt ds dt R 

where the third form is got by using (72). This shows the 
rate of acceleration perpendicular to the motion. It is towards 
the centre of curvature, of amount -y 2 /R, the well known 

Tortuosity of a Curve, and Various Forms of Expansion. 

118. Again referring to Fig. 7, if the curve be not a plane 
curve, or be a tortuous curve, the osculating plane undergoes 
change as we pass along the curve. It turns round the tan- 
gent, and the measure of the tortuosity is the amount of turn- 
ing per unit step along the curve. It is d<f>/ds, if d<J> is the 
angle between the two osculating planes at the extremities of 
ds. It is equivalently denoted by the tensor of dN/ds, if N be 
the unit normal to the osculating plane. This is similar to 
the equivalence of dd/ds and the tensor of dT/ds in measuring 
the curvature before noticed, the equivalent plane in that case 
being the plane normal to T. 

The three vectors, T, N, and R lf form a unit rectangular 
system, for we have 

N = VTR 1 , ...... (76) 

since T and R x are perpendicular unit vectors. As we move 
along the curve this system of axes moves as a rigid body, since 
there is no relative change. They may, in fact, be imagined 
to be three perpendicular axes fixed in a rigid body moving 


along the curve in such a way that these axes keep always 
coincident with the T, R x , and N of the curve. As it moves, 
the body rotates. It rotates about the N axis only when the 
curve is plane, but about the T axis as well when tortuous. 
The vector change (infinitesimal) in T is perpendicular to it, in 
the osculating plane that is, the plane whose normal is N, 
and is directed towards the centre of curvature. And the 
vector change (infinitesimal) in N is perpendicular to it, also in 
the osculating plane, and directed towards the centre of curva- 
ture, the vector R from the curve to the centre of curvature 
being on the intersection of the two planes mentioned. 

Various expressions for the tortuosity may be obtained. Let 
it be Y the tensor of Y, given by 

' -< 77) 

Then, by (76) and the context, (77) gives one expression. A 
second is got by performing the differentiation, giving 

i . . . (78) 
ds ds 

But the first term on the right is zero, by (72), the two vectors 
after V being parallel. So 

Y = VT^i ..... (79) 
gives another expression. Or 

Y = V T^, ..... (80) 

considering the tensor only. But, since Y is parallel to R u we 
have YRj = Y ; therefore, by (79), 

- 1 . . (81) 


is a third and entirely different expression. That Y is parallel 
to R! we may prove by (77) and (79). Multiply (77) by N. 

Then YN = N^ = j4N 2 =0, . (82) 

ds ds 

since N 2 = l ....... (83) 

This shows Y is perpendicular to N. It is also, by (79), per- 
pendicular to T. So its direction is that of R r 


By the parallelepipedal property (40) we may also write (81) 


Here again, we may write 

R, dE, I dR, , ^ d 1 /Q ~ X 

L = -- = -- + K - .... (oo) 

ds ds-R R ds dsR 

Substituting this compound vector in (84), we see that the 
first of the resulting vector products vanishes because RJ and 
R" 1 are parallel. There is then left 

-^-L ..... (8G) 
ds R 

f, . . . (37) 
s as* 

by (72), or, in terms of r and derivatives, 

which is a known symmetrical form, but whose meaning is less 
easy to understand than several preceding expressions. 

Since, by the definition of a vector product we have V^b 
= 6 when B,^ and b are perpendicular, a x being a unit and b any 
vector, we may, from the first equation (77) conclude that 

, ..... (89) 

which is equivalent to a form given by Thomson and Tait. 

Similarly, Y = V T^, ..... (90) 


Y being perpendicular to T as well as to N. 

The reader may, perhaps, find the above relating to tortuosity 
hard to follow, though the previous matter relating to tangency, 
curvature, velocity, and acceleration may be sufficiently plain. 
The hardness lies in the intrinsic nature of tortuosity, as to 
which see works on analytical geometry. But the reader who 
wishes to get a sound working knowledge of vectors should go 
through the ordinary Cartesian investigations, and turn into 


vectors. Also, convert the above to Cartesian form through 
i, j, k, and verify all the results, or correct them, as the case 
may be. Space is too short here for much detail. 

Hamilton's Finite Differentials Inconvenient and Unnecessary. 

119. It is now desirable to say a few words regarding the 
method of treating vector differentiation employed by Hamilton, 
and followed by Tait. The latter speaks in his treatise (chapter 
L, 33, third edition) of the novel difficulties that arise in 
quaternion differentiation ; and remarks that it is a striking 
circumstance, when we consider the way in which Newton's 
original methods in the differential calculus have been decried, 
to find that Hamilton was obliged to employ them, and not the 
more modern forms, in order to overcome the characteristic 
difficulties of quaternion differentiation. (For the word 
"quaternion," we may read vector or vectorial here, because a 
vector is considered by Hamilton and Tait to be a quaternion, 
or is often counted as one. This practice is sometimes confusing. 
Thus the important physical operator y is called a quaternion 
operator. It is really a vector. It is as unfair to call a vector 
a quaternion as to call a man a quadruped ; although, four 
including two, the quadruped might be held (in the matter of 
legs) to include the biped, or, indeed, the triped, which 
would be more analogous to a vector. It is also often 
inconvenient that the name of the science, viz., Quater- 
nions, should be a mere repetition of the name of the 
operator. There is some gain in clearness by preserving the 
name " quaternion " for the real quaternion the quadruped, 
that is to say.) The matter is illustrated by the motion of a 
point along a curve. But it does not appear, from this 
illustration, where the obligation to depart froni common usage 
comes in. 

The subject is, however, returned to more generally in 
Chapter IV., on Differentiation, where we are informed that 
we " require to employ a definition of a differential somewhat 
different from the ordinary one, but coinciding with it when 
applied to functions of mere scalar variables." Here again, 
however, a most searching examination fails to show me the 
necessity of the requirement ; or, at least, that the necessity is 


To examine this matter, consider for simplicity a function 
u of a single scalar variable a and a single vector variable r, 


Following the common method of infinitesimal differentials, we 

du=f(r + dr,a + da)-f(r,a). ... (91) 

That is, on giving the infinitesimal increments dx and da to r 
and a we produce the infinitesimal increment du in the function. 
Here du, di and da are the differentials, and it is important to 
remark, in connection with the following, that they are infini- 
tesimal, and that du is really the increment corresponding to 
dr and da. 

Now, Hamiltonian differentials have a different meaning. 
Though they would be denoted by du, di and da, yet, being 
different, we shall here dignify them with brackets, and denote 
them by (du), (di), and (da), to distinguish them from the 
differentials in (91). The Hamiltonian differential (du) is then 
defined by 

(du) - [/(r + (>, + to) -/(r, )], . (92) 

on the understanding that n is infinity, and that (on which 
stress is laid) the differentials (di) and (da) are finite and 
perfectly arbitrary. 

That this process is circuitous is obvious, but is it necessary ? 
Let us examine into its meaning, and compare (92) with (91). 
First, divide (92) by n. We then have 

+ (>-/(,, a). . . (93) 

Now, we see at a glance, that (93) and (91) become iden- 
tical if 

*>-&. *-(*) *,-<*> (94) 

n n n 

The division of the finite differentials (da) and (di) by n pro- 
duces infinitesimal results, and the finite differential (du) is 
similarly reduced to an infinitesimal. Why then not employ 
the infinitesimal differentials at once, and avoid the circuitous 


When we proceed to form a differential coefficient, as du/dt, 
if r and a are functions of t, or du/da, if r is a function of a, 
we see that their values are equivalently expressed by (du)/(dt) 
and (du)l(da), using the finite differentials; because this merely 
says that the value of the fraction du/dt is unaffected when 
the numerator and denominator are each multiplied by the same 

In the above, the reasoning is the same, whether u is a scalar 
or a vector function of r and a. 

But whilst the differential coefficient comes out the same, yet 
the differential (du) is not the increment in u belonging to the 
finite increments (di) and (da\ nor is it anything like it, excep- 
tions excepted. 

The above is as fair and as clear a statement as I can make 
of the difference between common and finite differentials as 
applied to vectors ; and we see that the Hamiltonian plan is 
only a roundabout and rather confusing way of expressing what 
is done instantly with infinitesimal differentials. But then 
the Hamiltonian plan is said to be obligatory. There's the 
rub ! I can only give my own personal experience. After 
muddling my way somehow through the lamentable quater- 
nionic Chapter II., the third chapter was tolerably easy ; but 
at the fourth I stuck again, on this very matter of the obliga- 
tory nature of the finite differentials. I am no wiser now about 
it than I was then. Perhaps there is some very profound and 
occult mystery involved which cannot be revealed to the vulgar 
in a treatise ; Hamilton is a name to conjure by. Or perhaps 
I " began the study of quaternions too soon !" Or perhaps it 
is only a fad, after all. Anyway, I am willing to learn, and if 
I live long enough, may reach a suitable age. At present let 
us return to common infinitesimal differentials. 

Determination of Possibility of Existence of Differential 

120. Although we may employ the method of differentials 
in any equation involving vectors, using the symbol d, and 
although we may turn it into an equation of differential coeffi- 
cients by changing d to the scalar differentiator d/dt (for 
example), as in 116, yet we cannot usually turn differentials 
into differential coefficients when the variable is a vector. 


This is not because we cannot form the expressions du\dt or 
dH/di, but because they are not differential coefficients when 
the denominators are vector differentials. This will be easily 
seen from typical examples. (The failure has nothing to do 
with the previous question of finiteness or otherwise of 
differentials.) Let 

w = ab ........ (95) 

and let the vector b vary/ Then, 

...... (96) 

and, therefore, = (dT>)- l (B,db) = (db) l [a,(db) 1 l . . (97) 

where (db) l means the unit db. Here the imitation differential 
coefficient du/db is a fraud, because its expression on the right 
side of (97) is not clear of the differential db, whose direction 
it involves. Nor is equation (97) of any particular use, com- 
pared with (96). 

Again, in the vector equation 

R = Vab ...... (98) 

let b be variable. Here we have 

..... (99) 

and from this equation of differentials we may form dH/db and 
VdR/db ; but neither of them, nor any combination of them, is 
a differential coefficient clear of differentials. Thus, 

. . (100) 
contains the differential on the right side. And 


also contains the differential. As a special result the value of 
(100) is zero (a parallelepiped in a plane) : and by equation 
(52) we may develope (101) to 

(rfb^a^b^-a ..... (102) 

But it is (99) that is the really useful fundamental equation, 
the developments (100) to (102) being useless for ditferentiating 
purposes, although true and interpretable geometrically, 


Generally, if 

.... (103) 
is a function of 


then Rj, R 2 , R 3 are functions, usually different, of x, y and 2, 
and the ratio of dR to di (even in the most general quater- 
nionic sense), cannot possibly be freed of the differentials dx, 
dy, dz. 

Even when we reduce the generality of the problem by 
making R and r coplanar, say in the plane of i, j, with z = 
and R 3 = 0, dR/dr is not usually a definite differential co- 
efficient. It only becomes one by special relations between 
the differential coefficients of Rj and R 2 with respect to x and y, 

=__^52. . . (105) 

^ __ 

dx dy ' dy dx 

This is the vectorial basis of the large and important branch 
of modern mathematics called the Theory of Functions, which 
appears in electric and magnetic bidimensional problems. 

Variation of the Size and Ort of a Vector. 

121. Sometimes it is convenient to have the variation of a 
vector exhibited in terms of the variations of its tensor and 
unit vector, or its size and ort. Thus, from 

we produce, by differentiating, 

dR = EdE l +H l dU ..... (106) 
If there be no variation of direction, then 

^R = R^R; ..... (107) 

and if the tensor does not vary, whilst the direction does, then 
dH = ~RdR l ..... . (108) 

Dividing (107) by the scalar c?R, we get a proper differential 

= E ...... < 109 > 


provided dR 1 = 0. From (108), 

provided dE = 0, which makes c?R perpendicular to E. But it 
is as well to avoid forms of this kind (with a vector variation in 
the denominator), even when by some limitation, as above, we 
can get rid of the differential. Of course, in (106), without 
any limitation, we can change d to d/dt (or other scalar diffe- 
rentiator) ; thus, 

R = RR 1 + R 1 R ...... (Ill) 

It is this (scalar) kind of differentiation that nearly always 
occurs in physical applications, and there is no abstrusity about 
it. See 116. Nor is there about differentials, provided we 
do not attempt to make differential coefficients with respect to 
vectors, or make the differentials themselves finite. 

We have also R 2 = R 2 , 

which, on differentiation, gives 

RrfR = RdR; ..... (112) 
or, dividing by R, 

..... (113) 

which says that the increment in the tensor is the component 
along the vector of its vector change dR, whose full expression is 
given by (106). Using (106) in (113) we get 

= ....... (114) 

Using (113) in (106) we get 

JR = RdK 1 + R 1 (R 1 ^),. . . . (115) 
or, dividing by R, 

^ = ( /R 1 + ^5, ..... (116) 

it xx. 

all of which results may be geometrically interpreted. 

In differentiating the reciprocal of a vector, proceed thus : 

,< m) 

where, for d may be written djdt, <kc. 


Another way is 

It may easily be shown, by (106), that these results are 

Enough has now been said about the meaning and effect of 
differentiating vectors. The important vector differentiator V 
deserves and demands a separate treatment. 

Preliminary on V- Axial Differentiation. Differentiation 
referred to Moving Matter. 

122. The Hamiltonian vector V occurs in all physical 
mathematics involving three dimensions, when treated vec- 
torially. It is denned by 

V = i-^+j/+k^, .... (119) 
dx dy dz 

the second form being merely an equivalent one, often more 
convenient, and easier to set up. We see that the Hamiltonian 
is a fictitious vector, inasmuch as the tensors of its components 
are not magnitudes, but are differentiators. As, however, 
these differentiators are scalar not scalar magnitudes, but 
scalar operators, having nothing vectorial about them the 
Hamiltonian, in virtue of i, j, k, behaves just like any other vec- 
tor, provided its differentiating functions are simultaneously 
attended to. Of course, an operand is always implied, which 
may be either scalar or vector. 

The Hamiltonian has been called Nabla, from its alleged 
resemblance to an Assyrian harp presumably, only the frame 
thereof is meant, without any means of evoking melody. On 
the other hand, that better known musical instrument, the 
Triangle, perhaps equally well resembles V> and- it does not 
want strings to play upon. 

First notice that 

iV = Vi, JV=-V 2 , kV = V 8 , - . (120) 
and that these are all special cases of the scalar product 

3 , . (121) 


where N is any unit vector. Here NV or its equivalent 
Cartesian expansion, means differentiation with respect to 
length, say s, measured along the axis of N, usually denoted 


ds dx dy dz 

or by similar forms. This operation was termed by Maxwell 
differentiation with respect to an axis, which we may more 
conveniently call axial differentiation. We see, therefore, that 
N Vj or the N component (scalar) of V> is an axial differentiator, 
the axis being that of N. Thus, 

= , . . (123) 
ds ds 

P being scalar and A vector. Also 

NV.AB=A.NV.B+B.NV.A, . . (124) 
NV.VAB=V(NV.A)B+VA.NV.B, . (125) 

and so on, simply because N V is a differentiator. Expanded in 
Cartesians, (124) contains the eighteen differential coefficients 
of the components of A and B with respect to x, y^ and z. The 
last two equations will also serve to illustrate notation. The 
dots are merely put in to act as separators, and keep the proper 
symbols connected. Mere blank spacing would do, but would 
be troublesome to work. We may, however, use brackets 

Thus, A.NV.B may be written A(NV)B, and this form may 
be used with advantage if found easier to read. Of course, N V 
being scalar, on its insertion between the two members of AB, 
the result remains a scalar product. Similarly, VA.NV.B, 
arising from VAB, is still a vector product. 

A somewhat more general operator is vVj where v is any 
vector. It is plainly v times the corresponding axial differen- 
tiator, or / y.v 1 V- Thus, expanded, 

vV-D = KVi + v a V 2 + %V 3 ) (iDi + JD 2 + kD 8 ), . (126) 

which we may separate into i, j, k terms if we like. 

When v is the velocity of moving matter, vV comes frequently 
into use. Let, for example, w denote the measure of some 



property of the moving matter (here a scalar, but it may equally 
well be a vector), and, therefore, a function of position and of 
time. Its rate of change with the time at a fixed point 
in space is dw/dt, and the matter to which this refers is changing. 
But if we wish to know the rate of time-change of w for the 
same portion of matter, we must go thus. Let Swfit denote 
the result, then 

Sw __ dw dw dx dw dy dw dz 
~St~di ~dxdi ~dydi ~ 

by elementary calculus. Or 

where v is the velocity of the matter. 

This is the equation extensively employed in hydrokinetics. 
In elastic solid theory the term vV- w is generally omitted, 
being often small compared with the first term on the right. 
But, in special applications, it may happen that any one of the 
three terms in (127) vanishes. Thus the term on the left side 
obviously vanishes when w keeps the same in the same matter. 
If, however, for w we substitute a vector function, say A, even 
though it may not suffer change at first sight in the same 
matter, yet the rotational part of the motion will alter it by 
turning it round, so that SA/St will not vanish. 

The operator 8/8t is distributive, like dfdt and vV> a s 
exemplified in (124), (125). 

Motion of a Rigid Body. Resolution of a Spin into other 


123. If the moving matter is so connected that it moves 
as a rigid body, we have a further development. If we set a 
rigid body spinning about a fixed axis whose direction is defined 
by the unit vector a 1} with angular speed a, the speed of any 
point P in the body equals the product of its perpendicular 
distance from the axis into the angular speed, and the direction 
of P's motion is perpendicular to the plane through the axis 
containing P. That is, a given particle describes a circle round 
the axis. This is obviously necessitated by the rigid con- 


nection of the parts of the body (or detached bodies). The 
velocity of P is therefore expressed by 

v = Var, ..... (128) 

if a = aa 1 , and r is the vector to P from any point on the 
axis. This is by the definition of a vector product. 

Here we may observe a striking advantage possessed by the 
vectorial method. For (128), obtained by elementary consider- 
ations, proves without any more ado that angular velocities 
about different axes compound like displacements, translational 
velocities, or forces, or, iu short, like vectors. Thus, in (128) 
we see that the velocity of P is the vector product of a and r. 
The latter fixes the position of P. The former depends on the 
spin, its tensor being the angular speed, and its direction that 
of the axis of spin. Now every vector, as before seen, may be 
expressed as the sum of others, according to the rule of vector 
addition. But every vector has its species. Thus, r being a 
vector distance, its components are vector distances. Similarly, 
a being a. vector axis of spin with tensor equal to the angular 
speed, the components of a, for instance, the three terms on 
the right of 

are vectors of the same nature. Therefore any spin may be 
replaced by other spins about other axes, according to the 
vector law. Of course the body cannot spin about more than 
one axis at once, but its motion is the same as if it could and 
did. This property is notoriously difficult to understand by 
Cartesian mathematics. 

The most general kind of motion the rigid body can possess 
is obtained by imposing upon the rotational any translational 
velocity common to all parts. Let q be this translational 
velocity, then, instead of (128), 

v = q + Var ...... (129) 

Here observe that q is the velocity of the point 0, or of any 
other point on the axis of rotation passing through the point 0. 
We may infer from this that if we shift the origin to Q, a 
point on a parallel axis, equation (129) will still be true, pro- 
vided r means the vector from the new origin Q to P, and q 
means the translational velocity of Q itself. Of course, the 


axis of rotation is now through Q. We may formally prove it 
thus. Take 

r = h + R, (130) 

where h is the vector from to Q and R the vector from Q 
to P. Put (130) in (129), then 

v = q + Vah + VaR (131) 

But here a + Vah is, by (129), the velocity of Q, say u, 
so we have 

v = u + VaR, (132) 

showing the velocity of P to be the sum of the translational 
velocity u at any point Q, plus the velocity at P due to rota- 
tion about the axis through Q. 

All motion is relative. But observe the absolute character 
of the spin a. It is absolute just because it involves and 
depends entirely upon relative motions. 

The translational velocity u of the point Q consists of a 
motion along the axis of rotation, combined with a transverse 
motion. The latter may be got rid of by shifting the axis. 
For if w is the velocity of Q transverse to the axis, if we go to . 
the distance w/a from the axis through Q, keeping in the plane 
through the axis perpendicular to w, we shall reach a place 
where the velocity due to the rotation about the Q axis is 
either + w or w, according to which side of the Q axis we 
go. Choosing the latter, where the transverse motion is can- 
celled, and transferring the axis to it. we see that the general 
motion of a rigid body consists of a spin about a certain 
axis (which is termed the central axis), combined with a 
translation along this axis. That is, it is a screw motion. It 
may, however, be more convenient not to employ the screw 
motion with a shifting axis, but to use (132), and let u and a 
be the translational and rotational velocities of a point Q 
fixed in the body. 

We may also have simultaneous spins about axes which do 
not meet. Thus, let R be the vector from any origin to a 
point Q about which there is a spin a, and r the vector from 
to any other point P. Then the velocity of P due to the 
spin is, by the preceding, 

v = Va(r-R) (133) 


Next let there be any number of points of the type Q, each 
with its spin vector a. The velocity at P is then the sum of 
terms of the type (133), in which r is the same for all. The 
result is therefore 

v = VAr-2VaR, .... (134) 

where A is the sum of the a's, or the spin of the body, whilst 
the minus summation expresses the velocity at 0. For example, 
two equal spins, a, in opposite senses about parallel axes at dis- 
tance h combine to make a mere translational velocity perpen- 
dicular to the plane of the axes; speed ah. I give these 
examples, in passing, to illustrate the working of vectors. 
These transformations are effected with a facility and a sim- 
plicity of ideas which put to shame the Cartesian processes. It 
may, indeed, be regarded as indicative of a mental deficiency 
to be unable to readily work the Cartesian processes which is 
in fact my own case. But that does not alter the fact that if 
a man is not skilful in Cartesians, he may get along very well 
in vectors, and that the skilful mathematician who can play 
with Cartesians of great complexity, could as easily do far 
more difficult work in vectors, if he would only get over the 
elements, and accustom himself to the vectors as he had to do 
to the other method. 

Motion of Systems of Displacement, &c. 

124. Going back to (127), we should, when the property 
whose time-variation is followed belongs to matter moving 
as a rigid body, employ in it the special reckoning of v of 
equation (132), giving the velocity of any point P in terms of 
u and a at an invariable point Q in the body. Thus, we have 

S = i!r (n+VaIl)v - w - (135) 

But equation (127) applies not merely to the case of matter 
moving through space, carrying some property with it, but also 
when the matter itself is fixed, whilst some measurable pro- 
perty or quality moves through the matter, or is transferred 
from one part of the matter to another. And here we may 
leave out the consideration of matter altogether, and think 
only of some stationary medium which can support and through 
which the phenomenon can be transferred. 


Thus, we may have a system of electric displacement D in a 
dielectric, which may be moving through it independently of 
any motion it may possess, and apply (127) to calculate D. It 
is, however, only in relatively simple cases that this can be 
followed up. For it is implied that we know the motion of the 
displacement and how it changes in itself, and these may be 
things to be found out. 

But should the displacement system move as a rigid body, 
the matter is greatly simplified. This will occur when the 
sources of the displacement (electrification, for instance) move 
as a rigid body, and the motion is of a steady type, so long 
continued that the displacement itself (in the moving system) 
is steady. Writing D for w in (135), we shall have 

~=VaD-(u + VaR)V.D .... (136) 

Here VaD is the rate of time-variation of the displacement in 
a moving element of the displacement system, caused by the 
rotation. It is zero when the displacement is parallel to the 
axis of spin, and a maximum when it points straight away 
from the axis, like the spoke of a wheel. 

The displacement system is, however, not necessarily or 
usually the same when in steady motion as when at rest. 
In the latter case only can it be regarded as known initially. 
Set it steadily moving and it will (by reason of the self- 
iuduction) be changed to another displacement system. Never- 
theless (136), along with the electromagnetic equations, enable 
us to find the new displacement system. 

Should, however, the velocities of the connected sources be 
only a very small fraction of the speed of propagation of dis- 
turbances through the medium, the re-adjustment of the dis- 
placement as the sources move takes place practically 
instantaneously (as if the speed of propagation were infinite), 
so that the displacement system remains unchanged ; preserving 
its stationary type, whilst it moves through the medium as 
a rigid body, in rigid connection with the sources. (This is 
obviously quite incorrect at a great distance from the sources, 
but there the effects themselves are insensible.) In this case, 
too, it is clearly unnecessary that the motion of the sources 
should be steady. Thus, in (136), D becomes the known 


displacement system (of equilibrium), moving as defined by u 
and a, which may vary anyhow. 

Similar considerations apply to systems of induction, moving 
with their sources. A general caution, however, is necessary 
when there are conductors or other bodies in the field, not 
containing sources. To keep them from having currents 
induced in them, or in other wys upsetting the regularity of 
the moving system, they should also partake in the motion, by 
rigid connection. 

Motion of a Strain-Figure. 

125. Equation (136) also applies to the motion through a 
medium of a " strain-figure," treated of by Dr. C. V. Burton 
in the current number of the Phil. Mag. (February, 1892). It 
is similar to that of slow motion of an electrical displacement 
system. Imagine a stationary infinite elastic medium with 
inertia, somehow set into a strained state, and let the speed of 
propagation of disturbances of strain be practically infinite. 
If then the strain-figure can move about through the medium, 
it will do so as a rigid body, provided the sources of the 
strain do so. In the above, D may signify the displacement 
(ordinary), whose variation in space constitutes the strain. 
Then D is the velocity of displacement, and is expressed in 
terms of the velocity of the strain-figure. From the expres- 
sion for the kinetic energy of the complete strain-figure, 
the mechanical forces concerned can be deduced. As this 
is to be done on dynamical principles involving Newton's 
laws, we may expect beforehand that a strain-figure sym- 
metrical with respect to a centre will behave as a New- 
tonian point-mass ; as does a similar electrical displacement 
figure. But, in general, without symmetry, the calcu- 
lation of the forces concerned would be very troublesome 

But the real difficulty appears to me to be rather of a 
physical than a mathematical nature. We have first to get 
some idea of how the strain-figure is kept up. Let it be 
stationary first. Then the strain-figure is referred to a 
forced or unnatural state of the medium in certain places. 
At any rate, we require intrinsic "sources" somewhere, and 
perhaps it might for this reason be convenient to consider 


these portions of the medium (with the sources) to constitute 
the atom or molecule, rather than the whole strain-figure. 

Now, having got a stationary strain-figure, how is it to be 
set moving? Three ways suggest themselves. First, a bodily 
motion of the medium carrying the strain-figure with it. This 
is plainly inadmissible. Next a motion of the atomic portions 
only of the medium (with the sources) through the rest of the 
medium, either disturbing it to some extent near by, or not 
disturbing it at all slipping through, so to speak. This 
would carry on the strain-figure. But it is inadmissible, if it 
be our object not to move the medium at all in any part. 
Thirdly, we may keep the whole medium at rest, and cause 
the sources themselves to move through it, so that the atomic 
portions of the medium change. But there is no means for 
doing this presented to our consideration. It lies beyond 
the dynamical question of the forces on an atom brought 
into play if the strain-figure can move in the manner sup- 
posed. The second course above, on the other hand, is more 
intelligible, although it implies something akin to liquidity. 

These objections are, however, only made suggestively. 
The matter is sufficiently important to be deserving of a 
thorough threshing out.* 

Space- Variation or Slope VP of a Scalar Function. 

126. The simplest and most easily understood effect of V 
upon a function is when it acts upon a scalar, say P. This 
implies that P is a scalar function of position, or of x, y> z. In 
the most important applications P is single-valued, as when it 
represents density, or pressure, or temperature, or electric 
potential, or the corresponding magnetic potential of magnets. 
But P may also be multiplex, as when it is the magnetic 
potential outside a linear electric current. At present let P be 

By (119) we have 

VP = LV 1 P+j.V 2 P + k.V 3 P. . . . (137) 

Of course, since v is vector and P scalar, the result is a vector. 
Its meaning is easily found. From the fact indicated in (137) 
that the rectangular scalar components of V P are the rates of 
increase of P along the axes of i, j, k, we may conclude that 

[* In case of eolotropy, add to the right member of (26) the term Sa, 
where S is .the torque and a the spin.] 


the component of vP in any direction is the rate of increase of 
P in that direction. Thus, N being any unit vector, we have, 
by (137), 

3 P. . (138) 

Comparing with (121), we see that 

. . . (139) 

by (123), s being length measured along N. 

The identity of NvP and Nv-P is tolerably obvious algebrai- 
cally. It is not, however, true when for P is substituted a 
vector, as we shall see later. 

If we take P = constant, we obtain the equation to a surface. 
If, then, T is any tangent to the surface at a chosen point, 
that is, any line in the tangent plane perpendicular to the 
normal, we shall have TvP = 0. The direction of vP itself 
is therefore that of the normal to the surface; or the lines of 
vP cut the equipotential surfaces perpendicularly, and pass 
from one to the next (infinitely close) by the shortest paths. 
The distance between any two consecutive equipotential sur- 
faces of a series having a common difference of potential varies 
inversely as the magnitude of vP, and vP itself is the vector 
showing at once the direction and the rate of the fastest 
increase of P. No perfectly satisfactory name has been found 
for this slope, or space- variation of a scalar function. Com- 
paring P with height above the level on a hillside, vP shows 
the greatest slope upwards. But the illustration is inadequate, 
since on the hillside we are confined to a surface. 

The tensor of vP is given by 

(VP)2 = (V 1 P) 2 + (V 2 P) 2 + (V 3 P) 2 , . . (140) 

as with any other vector. 

We may also here notice the vector product VNv in its 
effect on a scalar. We have, by the semi-Cartesian formula 
for a vector product, 

VNV = i(N 2 V 3 - N 3 V,) + j(N 8 V 1 - NiVg) + k(N x V 2 - N^). (141) 
Thus, when the operand is scalar, as P, we shall have 

(VNV)P = VNVP ...... (142) 


The result is a vector perpendicular to the plane of N and vP, 
*s in any other vector product, vanishing when they are 
parallel, and a maximum when they are perpendicular. 

Scalar Product VD. The Theorem of Divergence. 

127. When the operand of v is a vector, say D, we have 
both the scalar product and the vector product to consider. 
Taking the former alone first, we have 

divD = VD = V 1 D 1 + V 2 D 2 + V 3 D 3 . . . (143) 

This function of D is called its divergence, and is a very impor- 
tant function in physical mathematics. Its general signification 
will be best appreciated by a consideration of the Theorem of 

Let liquid be in motion. The continuity of existence of 
the matter imposes certain restrictions on the motion. The 
current is mq per unit area, whore m is the density and q the 
velocity. But, to further simplify, let m = 1, making the liquid 
incompressible. Then the current is simply q, which measures 
the amount of liquid crossing unit. area of any surface per- 
pendicular to q, per second. But if the surface be not perpen- 
dicular to q, the effective flux is only Nq, the normal component 
of q, if N denotes the unit normal to the surface. Therefore, 
2 Nq, or the summation of Nq over any surface, expresses the 
total flux of liquid through the surface. 

Now suppose the surface (fixed in space) is closed. Then the 
summation represents the amount of liquid leaving the enclosed 
region per second through its boundary, if N be the outward 
normal. This amount is evidently zero, because of the assumed 
incompressibility. If then we observe, or state, that 2 Nq is 
not zero, either the fluid is compressible, or else there must be 
sources of liquid within the region. Adopting the latter idea, 
because it simplifies the reasoning, we see that the summation 
2 Nq is the appropriate measure of the total strength of the 
sources in the region, since it is the rate at which liquid is 
being generated therein, or the rate of supply. The position of 
the internal sources is quite immaterial, and so is. the shape of 
the region, and its size, or the manner in which a source sends 
out its liquid (i.e., equably or not in all directions). So we have 


if p represents the strength of a source, and the 2 includes 
them all. 

When the sources are distributed continuously, so that p is 
a continuous function of position, its appropriate reckoning is 
per unit volume. That this is the divergence of q is clear 
enough, according to the explanations relating to divergence 
already given, ( 51). That it is the same as vq, as in (143), we 
may prove at once by the Cartesian form there exhibited. 

We have to reckon the flux outward through the sides of a 
unit cube. Take its edges parallel to i, j, k respectively. Then 
there are two sides whose outward normals are - i and i, and 
their distance apart is unity. The outward fluxes due to them 
are therefore - iq, or - q v and q l + v^, whose sum is v^. 
Similarly the two sides whose normals are j contribute 
V 2 g 2 , and the remaining sides contribute V 3 ^ 3 . Comparing 
with (143), we verify the Cartesian form of the divergence of 
a vector. 

But we may have any number of other special forms, accord- 
ing to the co-ordinates we may choose to employ for calculating 
purposes, such as spherical, columnar, &c., and the most ready 
way to find the corresponding form is by the immediate appli- 
cation of the idea of divergence to the volume-element concerned. 
For purposes of reasoning, however, it is best to entirely 
eliminate the idea of co-ordinates. Divergence is independent 
of co-ordinates. 

The sources need not be so distributed as to give rise to a finite 
volume-density. We may have, within the region concerned, 
surface-, line-, or point-sources, and the principle concerned 
is the same throughout. Thus the density of a surface-source 
is measured by the sum of the normal fluxes on its two sides 
per unit area, that is, by 2 Nq applied to the two sides of the 
unit area, if the flux wholly proceeds outwards. This is, 
however, not fully general, as there may be a flux in the 
surface itself, so that the full measure of the surface-density 
has to include the divergence of the surface-flux, to be found by 
calculating the flux leaving the unit area across its bounding 
line. Similar considerations apply to linear sources. In the 
case of a point-source the measure of the strength is the flux 
outward through a closed surface enclosing the point infinitely 
near it the surface of the point, so to speak. Any other- 


surface enclosing the point will do, provided there are no other 
sources brought in by the change. There may also be multiplex 
sources. Thus, a pair of equal point-sources, one a source, the 
other a sink, would be equivalent to no source at all if brought 
infinitely near one another ; but if the reduction in distance be 
accompanied by a corresponding increase in strength of the 
sources, the final result is not zero. This is the case of a 
magnetised particle on the theory of magnetic matter ; but it 
is not necessary or desirable to complicate matters by entering 
upon special peculiarities of discontinuity in considering the 
Divergence Theorem. Its general form is 

2ND = 2divD, (145) 

when the vector D, to which it is applied, admits of finite 
differentiation ; and the special meanings to be attached to the 
divergence of D, in order to satisfy the principle concerned, 
may be understood. 

Although a material analogy, as above, is very useful, it is 
not necessary. Any distributed vector magnitude will have 
the same peculiarities of divergence as the flux of a liquid. If 
it be the motion of a real expansible fluid that is in question, 
then the divergence of its velocity represents the rate of ex- 
pansion. It would, however, be very inconvenient to have to 
carry out this analogy in electric or magnetic applications ; 
an incompressible liquid, with sources and sinks to take the 
place of expansions and contractions, is far more manageable. 

Extension of the Theorem of Divergence. 

128. The following way of viewing the Divergence Theorem, 
apart from material analogies, is important. Consider the 
summation 2 ND of the normal component of a vector D over 
any closed surface. Divide the region enclosed into two 
regions, Their bounding surfaces have a portion in common. 
If, then, we sum up the quantity ND for both regions (over 
their boundaries, of course), the result will be the original 
2 ND for the complete region. The normal is always to be 
reckoned positive outwards from a region, so that on the 
surface common to the two smaller regions N is + for one 
and - for the other region, and 2 ND for one is the negative of 
that for the other, so far as the common surface goes. 


Since this process of division may be carried on indefinitely, 
we see that the summation 2ND for the boundary of any region 
equals the sum of the similar summations applied to the surfaces 
of all the elementary regions into which we may divide the 
original. That is, 


where, on the right side, we have a volume-summation whose 
elementary part <(D) is the same quantity 2ND as before, 
belonging now, however, to the elementary volume in question. 
We have already identified <(D) with the divergence of D. 

But if this demonstration be examined, it will be seen that 
the validity of the process whereby we pass from a surface- to a 
volume-summation, depends solely upon the quantity summed 
up, viz., ND, changing its sign with N. We may therefore at 
once give to the Divergence Theorem a wide extension, making 
it, instead of (146), take this form : 

2F(N) = 2/(N) ..... (147) 

Here, on the left side, we have a surface-, and on the right side 
a volume- summation. The function F(N), where N is the out 
ward normal, is any function which changes sign with N. The 
other function /(N), the element of the volume-summation, is 
the value of 2F(N) for the surface of the element of volume. 
Thus, by considering a cubical element, 

/(N)=V 1 F(i) + V 2 F(j) + V 3 F(k) . . (148) 

is the Cartesian form, should F(N) be a scalar function, or the 
semi-Cartesian form should it be a vector function. A few 
examples of the general theorem (147) will be given later. In 
the meantime the other effect of v should be considered. 

Vector Product WE, or the Curl of a Vector. The Theorem 
of Version, and its Extension. 

129. The vector product of v and a real vector, say E, is 
given in semi-Cartesian form by 

WE = i(V 2 E 3 - V 3 E 2 ) + j^ - 

= curlE. 


As before, with respect to the divergence of a vector, we can 
best appreciate the significance of this formula by the general 
property involved, expressed by the Theorem of Version. 

On any surface draw a closed curve or circuit. Let, for 
distinctness, E be electric force. Calculate the voltage in the 
circuit due to E. The effective force per unit length of circuit 
is the tangential component of E, or TE, if T is the unit 
tangent. The voltage in the circuit is, therefore, 2 TE, the 
summation being circuital, or a line-integration extended once 
round (along) the closed curve. 

Now draw on the surface a line joining any two points of 
the circuit. Two circuits are thus made, having a portion in 
common. Reckon up the voltage in each of the smaller circuits 
and add them together. The result is the voltage in the 
first circuit, if we rotate the same way in both the smaller 
circuits as in the original, because the common portion 
contributes voltage equally and oppositely to the two smaller 

This process may be carried on to any extent by drawing 
fresh lines on the surface. We therefore have the result that 
the voltage in the circuit bounding a surface equals the sum of 
the voltages in the elementary circuits bounding the elements 
of surface. Or 

2TE = 20(E), (150) 

where on the left side we have a circuital summation, and on 
the right side an equivalent surface-summation, in which 0(E), 
the quantity summed, is the value of 2 TE, that is, the voltage, 
in the circuit bounding the particular element of surface con- 

To find the form of in terms of i, j, k, &c., we need only 
calculate the voltages in unit square elements of surface taken 
successively with edges parallel to j, k, to k, i, and to i, j. In 
the first case, when the normal to the square circuit, or the axis 
of the circuit, is i, the voltage is 

V 2 E 3 -V 3 E 2 , 

that is, by (149), the i component of VvE, and therefore the 
normal component, since here N = i. Similarly when N=j, and 
the axis of the circuit is j, the voltage in it is expressed bv the 


coefficient ofj in (149). And when N = k, the voltage is the 
coefficient of k in (149). Thus, in any case, the voltage in an 
elementary circuit of unit area is the normal component of 
VvE, that is, NVvE. So (150) becomes 

, . . (151) 

expressing the Theorem of Version, sometimes termed Stoke's 

When E is not electric force, 2 TE is not circuital voltage, 
but circuital something else ; but this does not affect the 
general application of the theorem. 

This theorem is particularly important in electromagnetism 
because it is involved in the two fundamental laws thereof, 
what were termed the First and Second Circuital Laws, or 
Laws of Circuitation, connecting together the electric and 
magnetic forces and their time-variations. (See 33 to 36, 
Chap. II., and later.) 

If the vector whose curl is taken be velocity q in a moving 
fluid, then curl q represents twice the spin or vector angular 
velocity of the fluid immediately surrounding the point in 
question ; its direction being that of the axis of rotation, and 
magnitude twice the rate of rotation. But I have not made 
use of the fluid analogy in describing and proving the Version 
Theorem, because it is not of material assistance. 

Since the validity of the process whereby we pass from a 
circuital summation of the tangential component of a vector to 
an equivalent surface-summation depends upon the fact that 
TE changes sign with T, we may generalise the theorem thus : 

2F(T) = 2/(T), ..... (152) 

where on the left we have a circuital and on the right a surface- 
summation, and F(T) is such a function of T as to change 
sign with T ; whilst /(T), the element of the surface-summa- 
tion, is the value of the former 2 F(T) for the particular element 
of surface in question. Of this general theorem a few examples 
will be given later. 

A few words regarding v, div and curl, terminologically con- 
sidered, may be useful. Since divergence and curl are expres- 
sible in terms of vex (a provisional name for v, which has been 
suggested to me), why not use the vex operator only, like 



the quaternionists ? The reasons, which are weighty, should be 

In the first place, we require a convenient language for de- 
scribing or referring to processes and results, expressing approxi- 
mately their essential meaning without being too mathematical. 
Now V alone is not convenient for this purpose. The scalar 
product of v and D conveys no such distinct idea as does 
divergence ; nor does the vector product of v and E speak so 
plainly as the curl or rotation of E. 

Besides, the three results of V, exemplified in vP, and vD, 
and VvE, are so remarkably different in their algebraical 
development and in their meaning, that it is desirable, even in 
the algebra, to very distinctly separate them in representation. 
Therefore, in the preceding part of the present work (as in all 
former papers), the symbol V is only prefixed to a scalar, as in 
V P, the space variation of P, whilst for the scalar and vector 
product are employed div and curl, in the formulae as well as 
in descriptive matter. 

There are, however, cases when it may be desirable to use v 
and Vv applied to vectors in formulae, namely, when the 
combinations of symbols are not so simple that their meaning 
and effect can be readily seen, and when it is required to 
perform transformations also not readily recognisable. The 
utility of V in its vectorial significance then becomes apparent, 
for one may use it alone, temporarily if desired, and work it as 
a vector, remembering, however, its other functions. Of this, 
too, some examples should be given. 

Five Examples of the Operation of V in Transforming from 
Surface to Volume Summations. 

130. Returning to the theorem (147), or 

2/, ...".. (153) 

let us take a few of the simplest cases that present themselves. 
Given any odd function F of N, the normal outwards from a 
closed surface over which the summation on the left side ex- 
tends, we convert it to an equivalent summation throughout 
the enclosed region by making/, the quantity summed, be the 


value of 2 F(N) for the surface of the element of volume. This 
last is most conveniently a unit cube, so that 

/=V 1 F(i) + V 2 F(j) + V 3 F(k),. . . (154) 
as in (148). 

(a). The simplest case of all is F(N) = N itself. Then, by 
(154), or by considering that the vector normals to the six faces 
of a cube balance one another in pairs, we have 

2N = 0, ...... (155) 

expressing that a closed surface has no resultant orientation ; 
or, that a normal pull applied to every part of a closed surface, 
of uniform amount per unit area, has no resultant. 

(b). If we multiply by p, any scalar function of position, we 
have the same case again if p be constant. When negative, it 
makes a well-known elementary hydrostatic result. But when 
p is not constant, then, by (154), 

Vjp, . . . (156) 
so that we have 

2Np = 2V# ...... (157) 

These are, of course, vector summations. The sum of the 
surface tractions equals the sum of the bodily forces vp arising 
from the space- variation of the internal tension. Take p nega- 
tive to indicate pressure. 

(c). Take F(N) = ND, where D is a vector function of position. 
This gives the most valuable theorem of divergence, 

= 2divD, . . . (158) 
already discussed. 

(d). Take F(N) = NDP, where P is a scalar. Here, by (154), 
or by (158), we shall find 

2NDP = 2V(DP) ...... (159) 

Here V has to differentiate both D and P, thus, 

, . . . . (160) 


so that the previous equation may be written 

. . . (161) 

which is a form of Green's Theorem relating to electrostatic 
energy. D may be the displacement in one system of electrifi- 
cation and P the potential in another. The quantity - 2DvP is 
their mutual energy ; and this is, by (161), equivalently ex- 
pressed by the sum of products of every charge in one system 
into the potential due to the other. 

(e). Take F(N) = VNH. Then, by (154), 

/= ViViH + V 2 VjH + V 3 VkH. 

But here we may shift the v's to the other side of the V's a 
because they are scalars ; this produces 

so that we have 

2VNH = 2VVH = 2curlH. . . . (162) 

The interpretation may be more readily perceived by reversing 
the direction of the normal. Take N = - n, so that n is the 
normal drawn inward from the boundary. Then 

2curlH + 2VnH = ..... (163) 

If H be magnetic force, curl H is the .electric current-density in 
the region. Now VnH is the surface equivalent of the bodily 
curlH. Ignore altogether the magnetic force outside the 
region, if there be any. Then the circuitation of H gives the 
current through a circuit. Applied to elementary circuits 
wholly within the region, the result is curl H. But at the 
boundary, where H suddenly ceases, there is a surface-current 
as well. To find its expression, apply the process of circuitation 
to a circuit consisting of two parallel lines of unit length, 
infinitely close together, but on opposite sides of the boundary, 
joined by infinitely short cross-pieces. Only the unit line 
inside the region contributes anything to the circuitation ; and 
by taking it to coincide with H, so as to make the circuitation 
a maximum, we find that VnH represents the surface-density 
of current. So, if J be current, we have, by (163), 2 J = for 
any region, by itself. The surface-distribution and the volume- 


distribution of current are complementary ; that is, they are 
properly joined together to make up a circuital distribution. 

nVVH=-VVnH, .... (164) 

where on the left side we have the divergence (at the surface) 
of the internal current, and on the right the equal convergence 
of the surface-current. 

Five Examples of the Operation of V in Transforming from 
Circuital to Surface Summations. 

131. Next, take a few examples of the extended Theorem 
of Version (152), viz. : 

2F(T) = E/, .' . . . . (165) 

where now, on the left side, we have the circuital summation of 
an odd function of T, and on the right an equivalent surface- 
summation, whose elementary part / is the value of F(T) for 
the circuit bounding the element of surface. 

Taking three elements of surface to be unit squares, whose 
normals are i, j, k, we readily see that the corresponding /'s are 

v 2 F(k) - v 3 F(j), v.FW-v^k), vjfl-vf.. (166) 

By means of these we can see the special form assumed by / in 
any case. 

(a). Thus, take F(T) = T itself, the unit tangent. Then we 

2T = 0, ... . . . (167) 

merely expressing the fundamental property of adding vectors, 
that the sum of any vectors forming, when put end to end, a 
circuit, is zero. 

(6). Take F(T) = TP, where P is a scalar function of position. 
Then, with normal i, we have, by the first of (166), 

; . . (168) 
so, writing N for i, we obtain 

- . . (169) 


The quantity summed over the surface is, therefore, the 
surface representative of the curl of vP. This has no volume 
representative, its value being then zero. 

(c). Take F(T) = TH. Here, with normal i to the element 
of surface, we have 

/= V 2 kH - VgjH = V 2 H 3 - V 3 H 2 = iVVH, 
by (149). Therefore, putting N for i, 

2TH = 2NVVH = 2NcurlH . . (170) 

the Version Theorem again. But observe that, by the trans- 
formation (164), we may also write it 

2TH = 2VNV.H, . . . . (171) 

similarly to (169), in which the operand is a scalar. This is 
mnemonically useful, but (170) is more practically useful. 

(d). Take F(T) = VTH. Here, with normal i, the first of 
(166) gives 

/= V 2 VkH - V 3 VjH = V (kV 2 - JV 3 )H. 

But here we have kV 2 - j V 3 = ViV, 
so that /=V(ViV;H; 

and therefore, generally, putting N for i, 

2VTH = 2V(VNV)H. . . . (172) 

(e). Let the quantity in the circuital summation be a vector 
of length P (a scalar function of position) drawn perpendi- 
cularly to the plane of T and N. That is, 

F(T) = (VTN)P. 
We then find, taking N = i, and using the first of (166), 

/= V 2 Vki . P - V 3 Vji . P = (j V 2 + kV 3 )P = VP - i (iV) P. 
In general, therefore, 
2 VTN . P = 2 (VP - N . NVP) = 2 V a P = 2 V (VNV)N.P . (173) 

The element of the surface-summation is V 8 P, meaning the 
slope of P on the surface itself, disregarding any variation it 


may have out of the surface. The last form of (173) involves 

the transformation formula (52). 

* Observe that in all the above examples, 

2F(N) = 2F(V), (174) 

when we pass from a closed surface to the enclosed region; 
and that 

2F(T) = 2F(VNV), .... (175) 

when we pass from a circuit to the surface it bounds. Thus, 
N becomes V, and T becomes VNv. But I cannot recommend 
anyone to be satisfied with such condensed symbolism alone. 
It is much more instructive to go more into detail, as in the 
above examples, and see how the transformations occur, bearing 
in mind the elementary reasoning upon which the passage from 
one kind of summation to another is based ( 128, 129). 

Nine Examples of the Differentiating Effects of V 

132. The following examples relate principally to the modi- 
fications introduced by the differentiating functions of V. 

(a). We have, by the parallelepipedal property, 

NVVE = VVEN = EVNV, . . . (176) 

when V is a common vector. The equalities remain true when 
V is vex, provided we consistently employ the differentiating 
power in the three forms. Thus, the first form, expressing the 
N component of curl E, is not open to misconception. But in 
the second form, expressing the divergence of VEN, since N 
follows V, we must understand that N is supposed to remain 
constant. In the third form, again, the operand E precedes the 
differentiator. We must either, then, assume that y acts back- 
wards, or else, which is preferable, change the third form to 
VNV.E, the scalar product of VNv and E; or (VNv)E, if that 
be plainer. 

(b). Suppose, however, that both vectors in the vector pro- 
duct are variable. Thus, required the divergence of VEH, 
expanded vectorially. We have 

. . . (177} 


where the first form alone is entirely unambiguous. But we 
may use either of the others, provided the differentiating power 
of y is made to act on both E and H. But if we keep to the 
plainer and more usual convention that the operand is to 
follow the operator, then the third form, in which E alone is 
differentiated, gives one part of the result, whilst the second 
form, or rather, its equivalent -EVvH, wherein H alone is dif- 
ferentiated, gives the rest. So we have, complete, and with- 
out ambiguity, 

div VEH = H curl E-E curl H, . . . (178) 

a very important transformation. It is concerned in the de- 
duction of the equation of activity from the two circuital laws 
of electromagnetism. 

(c). In these circuital laws we have also to consider the curl 
of a vector product, viz., the curl of the motional electric force 
in one law, and the curl of the motional magnetic force in the 
other. Taking the former, we have 

curlVqB = VVVqB, . ... (179) 

where B is the induction and q the velocity of the medium 
supporting it. Apply the elementary transformation (52) to 
(179). It gives 

VVVqB = q.VB-B.Vq, ... (180) 

when y is a mere vector. But on the left side both q and B 
have to be differentiated ; therefore the same is true in both 
terms on the right side. This gives 

VVVqB = qdivB + BV.q 

-Bdivq-qV.B, . . . (181) 

without ambiguity or need of reservation. That is to say, as 
in the q. vB of (180) both q and B have to be differentiated, we 
get qdivB when B alone, and Bv.q when q alone is differ- 
entiated. Similarly for the other term in (180). 

Or we might write V q when q alone, and V B when B alone 
suffers differentiation. Then, fully, 

VVVqB = VV q VqB + VV B VqB, . . (182) 
VV q VqB = BV.q-Bdivq, . . (183) 

W B VqB = qdivB-qV.B. . . (184) 


Here the sum of (183) and (184) gives (181). The inean- 
iftg of By and qy has been already explained. 

(d). Equation (181) may be applied to the circuital laws. 
Take the second, for example, in the form (4), 66, 

- curl (E - e ) = K + B + wo- - curlVqB, . (185) 

and suppose that w = q, or that sources move with the medium. 
Then, by (181), we cancel the convective term wo-. Further, 

we have B + qy.B = SB/fo, b 7 ( 127 )> 122 so that ( 185 ) 

- curl (E -e ) = K + SB/8* + Bdivq-BV.q, . (186) 

and the corresponding form of the first law (equation (3), 
66), is 

curl(E-li ) = C + oD/o-* + Ddivq-Dy.q. . (187) 

The time-variations refer to the same (moving) portion of the 
medium now. But if we wish to indicate the movement of 
electrification, &c., through the medium, that is, have relative 
motion u - q of p (and w - q of <r) with respect to the ether 
then to the right side of (187) add the term (u-q)/>, and to 
the right ride of (186) add (w - q)<r. 

It is desirable to preserve the velocities u and w, or else the 
relative velocities, as well as the velocity q of the medium, in 
order to facilitate the construction and comprehension of 
problems relating to electromagnetic waves, which, although 
abstract and far removed from practice, are of a sufficiently 
simple nature to enable one to follow the course of events. 

(e). We have already had the divergence of the product of a 
scalar and vector under consideration. Now examine its curl. 

curlDP = VV(DP) 


= PcurlD-VDVP. . . . (188) 

Here V can only make a vector product with D, because P is 
scalar. On the other hand, both P and D suffer differentiation. 
So in the second line we have V both before and after D. 

(/). The divergence of the curl of any vector is zero. That 
is, divcurlH = 0, or VWH = 0. . - (189) 


If y here were a real vector (189) would mean that the 
volume of a parallelepiped vanishes when two edges coincide. 

(g). A somewhat similar case is presented by the vanishing 
of the curl of a polar force. Thus, 

curlVP = 0, or VV-VP = 0. . . . (190) 

Of course Vw is zero. But the scalar product VV, or v 2 , is 
the Laplacean operator, 

V2 = V 1 2 + V 2 2 + V 3 2 , .... (191) 
which occurs frequently. 

(h). Let the operation curl be done twice on a vector. Thus, 

(curl) 2 A = VVVVA 

= V-VA-V 2 A, . . . (192) 
by the transforming formula (52). Or 

V 2 A = VdivA-curl 2 A. . . . (193) 

Thus there are two principal forms. If the vector A has no 
curl, then V 2 A is the slope of its divergence. If, on the other 
hand, it has no divergence, then - V 2 has the same effect exactly 
as taking the curl twice. 

(i). In the case of the operand being a scalar, then we have 

V 2 P = divVP, (194) 

the divergence of the slope of the scalar. 

The Potential of a Scalar or Vector. The Characteristic 
Equation of a Potential, and its Solution. 

133. The last equation brings us to the theory of potentials. 
There are several senses in which the word potential has been 
employed, to enumerate which would be valueless here. For 
our present purpose we may conveniently fix its meaning by 
denning the potential at A of a quantity p at B to be the quantity 
/>/47ir, where r is the distance from B to A. This is the rational 
potential, of course. 

When p is distributed throughout space, whether at points, 
or over surfaces, or throughout volumes, the potential at any 


pofnt- is the sum of the potentials of all the elements of p. 
That is, 

, .... (195) 

if P is the potential of p. 

We may use the same definition when it is a vector that has 
to be potted, or potentialised. Thus, if A is the potential of 
0, then 

..... (196) 

The summation is now a vector summation. Also, pot means 
" potential," or " the potential of," and has no more to do with 
kettle than the trigonometrical sin has to do with the un- 
mentionable one. It seems unnecessary to say so, but one 
cannot be too particular. 

We may connect these potentials with v as follows : Given 

divF = />, ..... (197) 

that is to say, that the divergence of a vector F is p. The 
meaning of divergence has been explained more than once; 
both its intrinsic and its vectorial meaning. Now, if the vector 
F be explicitly given, it is clear that p is known definitely, 
since it is derived from F by differentiation, which should, 
perhaps, be regarded as a direct process, rather than inverse. 
But if it be p that is given, F is not immediately deter minable, 
unless we subject F to limitations. For we may construct any 
number of different F's to satisfy (197). Let every elementary 
source p of F send out the quantity p of F, according to my 
rational theory of sources already explained ; that is, the 
Unitarian system of one "line of force" to the unit "pole." 
Then, by the manner of construction, the resultant F will satisfy 
(197), and it will do so independently of the way we choose 
to let a source send out the flux it generates, whether equably 
or not. 

But if it be done equally in all directions, so that p/4nr 2 is 
the intensity of the " force " at distance r from the point-source 
p, and r 1 /o/47rr 2 the vector force to correspond, where r x is a 
unit vector drawn from p towards the point under considera- 
tion, making the resultant F be 

1 , ..... (198) 


we obtain a special solution of (197) which has a remarkable 
property, viz., 

curl F = 0, ..... (199; 

so that if F be electric force, the voltage in any circuit is zero. 
The meaning of curl, I may observe, has been explained more 
than once ; both its intrinsic and its vectorial meaning. Those 
who seek can find, If they will not take the trouble to seek 
or to remember it is of no consequence to them. There are 
plenty of other things they may concern themselves about ; 
perhaps more profitably. 

The property (199) is visibly true in the case of a single 
source. It is therefore separately true for the fields of all the 
sources, and therefore, by summation, is true for the com- 
plete F. 

But (198) does not give the only vector which has no curl 
and a given divergence. For a constant vector (that is, constant 
throughout all space), has no curl and no divergence, unless we 
go to the very end of space to find the sources. Of course this 
constant solution has no relation to the sources p, and may be 
wholly ignored. If allowed, F would not vanish at an infinite 
distance from the sources. Remembering this, and excluding 
the constant solution, we may say that (198) is the solution of 
(197) and (199). 

That there is no other solution may be proved analytically 
by Green's Theorem. But we do not really need any 
appeal to analysis of that kind, if the intrinsic meanings of 
divergence and curl are understood. For the admission that 
there could be a second solution, say, F + f, where F is the solu- 
tion (198;, would, by (197) and (199), imply that the vector 
f had no divergence, and also no curl anywhere. But the first 
of these conditions means that f is entirely circuital, if existent 
at all. The second denies that it is circuital. So f is non- 

Now observe that 

or the slope of the scalar l/47rr is the vector with tensor l/4?rr 2 
and direction T r It follows from this that 

-Vpotp = F ...... (200) 


when a single point-source is in question. Therefore, by sum- 
mation, the same is true for any distribution of sources, or 

F= -VP= -Vpotp, .... (201) 

where P is the potential of /o, as denned by (195), and F is as 
in (198). The slope of P, if by this we understand slope down- 
wards, or vector rate of fastest decrease, is therefore the same 
vector as was constructed to solve (197) subject to (199). 
Taking the divergence of (201), we have, by (197) and (194), 

/o = divF= -V 2 P= -V 2 pot/x . . . (202) 
A solution of the characteristic equation of P, or 

V 2 P=-/o, (203) 

is therefore (195), and it is the solution vanishing at an infinite 
distance from the sources. 

If we start from (203), we should first use (194), and make it 

div(-VP) = /> (204) 

Then, by (190), we see that - VP has no curl, so that we have 
again the two equations (197), (199) to consider, as above. 

The consideration of F rather than of P has many advan- 
tages for purposes of reasoning, as distinguished from calcula- 
tion. This is true even in statical problems; for instance, 
when F is electrostatic force, and P the corresponding potential. 
When we proceed further, to kinetic problems, when F can no 
longer be wholly expressed as the slope of a potential, the 
utility of considering P at all, even for calculating purposes, 
becomes sometimes very questionable, and the consideration is 
sometimes certainly useless and misleading. 

From (202) we see that - v 2 and pot are reciprocal. In 
another form, -V~ 2 and pot are equivalent; or (pot)" 1 and 
- v 2 are equivalent. The property has only been proved for a 
scalar function, having a scalar potential. But since any 
vector may be written iC 1 +jC 2 + kC 3 , and the property is 
true for the three scalars C lf &c., it is also true for the vector 
0. Thus, explicitly, 

- V 2 pot = - V 2 pot (iCi + jC 2 + kC 3 ) 
= i ( - V 2 pot CJ + j ( - V 2 pot CJ + k ( - V 2 pot C 3 ), 


because the reference vectors i, <fec., are constant vectors. So, 
if Aj is the potential of C 15 A 2 of C 2 , and A 3 of C 3 , which makes 
A be the potential of C, according to (196), we shall have 

A = potC, (205) 

-V 2 A = C, (206) 

-V 2 potC = C (207) 

In short, the characteristic equation of the vector A merely 
unites, from the above point of view, the characteristics of the 
components, so that pot and - V 2 are reciprocal when the 
operand is a vector, as well as when it is a scalar. 

Connections of Potential, Curl, Divergence, and Slope. Separa- 
tion of a Vector into Circuital and Divergent Parts. A 
Series of Circuital Vectors. 

134. But the above gives a very partial and imperfect view 
of the general theory of potentials. There are numerous other 
relations between a vector and its associated functions. For 
instance, if in (206), A be circuital, then the Laplacean y 2 
may, by (193), be replaced by -(curl) 2 . That is, if A x be cir- 
cuital, and be the potential of C 15 then, 

curl 2 A! = 0!, . . . . (208) 
or curl 2 pot C^Cj (209) 

Here, then, we have replaced the scalar operation - V 2 by the 
vector operation curl done twice. Of course, C x is also cir- 
cuital, as is proved by (189). 

Again, let the A of (206) be polar, or wholly divergent, and 
be now called A^ the potential of C 2 ; then, by (193), we shall 

-VdivA 2 = C 2 , . . . . (210) 

or -VdivpotC 2 = C 2 (211) 

Here again we have replaced V 2 by a double operation, first 
div and then V. This is similar to the passage from (203) to 
(204), only done in the reverse manner. In (210), by (190), 
C 2 is polar, because A 2 is. 

Conversely, we see that the potential of a circuital vector is also 
circuital, and that the potential of a polar vector is also polar. 


Now Aj has no divergence, so it may be added on to the 
divergent A 2 in (210) without affecting its truth. Thus, if 
A = Aj + A 2 , we have 

-VdivA = C 2 ..... (212) 

Similarly, A 2 has no curl, so may be added to the Aj in (208), 

cur! 2 A = C 1 ...... (213) 

Here remember that A is the potential of or Oj + 2 . 

These equations supply one way of effecting the division of a 
vector A of general type (having both curl and divergence) into 
two vectors, one of which, A I} is circuital, whilst the other A 2 
is polar. For A 2 is the potential of 0. 2 , so, by (212), 

-potVdivA = A 2 . . . . (214) 

separates A, from A. Similarly, A l is the potential of C 1? so 
by (213), 

pot curl 2 A = Aj .... (215) 

separates A x from A, and therefore A 2 from A by a different 
method. There are many other ways of splitting A into cir- 
cuital and divergent parts. The one most easily understood, 
apart from the mathematics, is the following. Go over the 
whole field of A and measure its divergence. If we find that 
there is no divergence, then we do not need to go further, for 
we know that A is circuital already; that is, A = A 1} and 
A 2 = 0. But should there be divergence, say B 2 , so that 

divA = divA 2 = B 2 , .... (216) 

then construct the flux A 2 corresponding to the divergence B 2 
according to the method already explained with respect to 
(197); thus, 

= -VpotB 2 , ..... (217) 

by (198) and (200). Knowing A 2 , we know A 1? or A - A 2 . 

Or we might vary the process thus. First measure the curl 
of A. This is the same as the curl of A l because the curl of 
A 2 is zero. Let, then, 

B 1 ..... (218) 


and construct the circuital solution of this equation ; that is to 
say, regarding B x as given, find A r It is given by 

A! = curl pot B! (219) 

For Aj as thus defined is evidently circuital, in the first place ; 
and next, by taking the curl, we produce 

curl AJ = curl 2 pot B^Bj, . , . (220) 

which is the given datum. Here we use curl 2 pot = l, because 
the operand is circuital, as in (209). 
But instead of (219) we may write 

A 1 =potcurlB 1 , . . . . (221) 

showing an entirely different way of going from B x to A r For 
A! as thus constructed is circuital ; and, since curl Bj = G v 
(221) is the same as 

A 1= pot Op 

which was our definition of A l in terms of C r 

Thus pot curl and curl pot are equivalent when the operand 
is circuital, as above. They are, however, also equivalent when 
the operand is general, or both circuital and divergent, because 
if any divergent vector be added to the B x in the right members 
of (219) or (221), the operation of curl to which it is subjected 
renders its introduction inoperative. We therefore have 

pot curl C = curl pot 0, .... (222) 

where is any vector. We have also the similar exchange- 

pot curl 2 C = curl 2 pot 0, .... (223) 

where C is any vector. For, either way, the result is the 
circuital part of C, or C r 

These results, though puzzling at first from their variety, 
are yet capable of being brought under rapid- mental control 
by bringing them together in a compact form. Thus, start 
with any circuital vector A r Let B x = curl Ap O x = curl Bp 
Dj =cur!0 1 , &c. We have a series of vectors 

Aj, B I} Op Dp Ep . . . 

which are all circuital, and any one of which is the eurl of the 


preceding. We thus pass down the series one step at a time 
by means of the operation of curling ; for example, 

D^curlOi (224) 

If, however, we wish to go down two steps, we do not need 
to go first one step, as above, and then another, also as above ; 
but can make a double step in one operation by means of the 
Laplacean v 2 . Thus, 

-VSB^Dj (225) 

Now go the other way. If we wish to rise up two steps, 
we can do it in one operation by potting ; thus, 

B^potDi (226) 

If we wish to go up only one step, we may do it by (224), 
(225), (226) combined ; that is, either go down one step first, 
and then up two, as in 

B 1 = pot Dj = pot curl G l - ) ... (227) 
or else, first go up two steps and down one j thus, 

B x = curl A! = curl pot O r . . . (228) 

There are other less important combinations. But if we 
wish to make one step up directly, without making use of the 
double step, we must do it by the Amperean formula, already 
used, whereby we pass direct from electric current to its 
magnetic force, which, in rational units, is (when applied to 
any pair of neighbours G l and D x in the above series), 

C 1 = 2(VD 1 r 1 )/47rr2, .... (229) 

where i^ is a unit vector from the element D 1 to the point at 
distance r therefrom, where G 1 is reckoned. We have now a 
complete scheme, so far as the circuital vectors are concerned. 

A Series of Divergent Vectors. 

135. Deferring temporarily a vectorial proof of the last 
formula (229), which is the only unproved formula in the con- 
nections of the series of circuital vectors, it will now be 
convenient to bring together the connections of the divergent 
vectors and associated quantities. We saw the advantage of 
the systematic arrangement of the connected circuital vectors 


to be like producing a harmonious chord out of apparently 
disconnected tones. The advantage is much greater in the 
divergent series, on account of the less uniform relations in- 
volved and the greater need of a system to bring them under 
rapid mental control. In the circuital series, four kinds of 
operation were involved ; but in the divergent series there are 
six. The chord will be found to be perfect, though of greater 
Thus, let 

A 2 , B 2 , C 2 , D 2 , E,, . . . . 

be a series of vectors and scalars connected as follows : Start 
with A 2 , which is to be any divergent vector ; that is, havinr 
no curl. Let B 2 be its divergence ; 2 the slope of B 2 ; D 2 
the divergence of 2 ; E 2 the slope of D 2 , &c. Then A 2 , C 2 , 
E 2 , . . . are all divergent vectors. But they are separated 
from one another by two steps instead of one, as was the case 
in the circuital series last treated. The intermediate quantities 
are scalars. Instead, also, of the single operation of curl 
which suffices, in the circuital series, to carry us from any 
vector to the following one, we now have two distinct opera- 
tions ; viz., that of slope, when we pass from a scalar to the 
next vector, as in 

C 2 =-VB 2 ; (230) 

and that of divergence, when we pass from a vector to the next 
scalar, as in 

D 2 = div0 2 (231) 

But if we wish to go down two steps at once, we can do so 
by means of the Laplacean operator, whether the operand be a 
scalar, as in 

-V 2 B 2 = D 2 , (232) 

or else a vector, as in 

-V 2 C 2 = E 2 . ...... (233) 

In this respect, then, we have the same property as in the 
circuital series. 

We have also identity of operation in going up two steps at 
once, whether from a scalar to the next higher scalar, as in 

B 2 = potD 2 , (234) 


or from a vector to the next higher vector, as in 

C 2 = potE 2 ; (235) 

any member of the series being the potential of the second 
after, as in the circuital series. 

Next, to go up one step only, we may utilise the preceding 
in two ways. First go up two steps and then down one, as in 

B 2 = divA 2 = divpotC2, . . . (236) 

where we pass from the vector C 2 to the scalar B 2 by pot 
first (up two steps), and then by div (down one step) ; and 
also as in 

2 = -VB 2 = -VpotD 2 , . . . (237) 

where we pass from the scalar D 2 to the vector 2 by pot first 
(up two steps), and then by - y (down one). 

Or, secondly, we may first go down one step and then up 
two, as in 

B 2 = potD 2 = potdiv0 2 , . . . (238) 

when rising from the vector C 2 to the scalar B 2 ; or as in 

2 = potE 2 = -potVD 2 , . . . (239) 

when rising from the scalar D 2 to the vector C 2 . 

Finally, if we wish to rise up one step at once, without using 
the double step either up or down, we can do it by means of 

C 2 = (div)- 1 D 2 = 2r 1 D 2 /47rr 2 , . . (240) 

when we rise from a scalar D 2 to a vector 2 , which is, in fact, 
the fundamental formula of the inverse- square law upon which 
our potential investigations are based. But in rising from a 
vector C 2 to the scalar B 2 just above it in the series, we require 
to use a different process, namely, 

B 2 = (-V)- 1 C 2 =-2r 1 2 /47rr2. . . (241) 

In these formulas (240), (241), r x is a unit vector from the 
element in the summation towards the place of the resultant ; 
that is, from D 2 to C 2 in (240), and from C 2 to B 2 in (241). 

We now have a complete scheme for the divergent vectors 
as we had before for the circuital series. On comparing them 
we see that they are alike in the double steps, either up or 
down, but differ in the single steps. There is but one kind of 



step up and but one kind down in the circuital series, whereas 
there are two kinds up and two kinds down in the divergent 
series. The down step in the circuital series is always done by 
curl; the up step, shown in (229), may be denoted by (curl)" 1 . 
In the divergent series the down steps are done by - v and by 
div; their inverses may be denoted by (-v)- 1 and (div)- 1 . 
It is now the nature of these inverse operations (229), (240), and 
(241) that remains to be elucidated vectorially. The first is 
the Amperean formula rationalized, whereby we rise from 
electric current to its magnetic force ; by the second we rise 
from (for instance) electrostatic force to the electrostatic 
potential ; or, with a slight change (of sign), from intensity of 
magnetisation to magnetic potential ; in the third we rise 
from (for instance), electrification to electrostatic displacement. 

The Operation inverse to Divergence. 

136. Let p and q be scalar functions, and consider the 
space-variation of their product. We have 

.... (242) 

a formula not previously used, but which is seen to be true 
by observing that it is true for each of the three components 
of V. Now integrate through any region. We know that 

2Np2 = SV(p2), ..... (243) 

if N is the unit normal outwards from the boundary of the 
region, so that the left member is a surface-summation, whilst 
the right member is a volume-summation throughout the 
region bounded by the surface. Equation (243) is, in fact, a 
case of (157), with pq substituted for p. If, then, the surface- 
summation vanishes, we shall have a simultaneous evanescence 
of the right member of (243), and therefore, by (242), 

2pVq=-2qVp. ...'.. (244) 

All the work done by a vector-analyst is exhibited in (244) 
itself, viz., the transfer of the symbol v from one operand 
to the other with change of sign, converts the integral of 
pvq into that of -gvp. The previous remarks contain the 
justification of the process, 


Now - vq is a polar or divergent vector, so may be any one 
of our divergent series, say C 2 , when q itself becomes B 2 . Then 

2^C 2 = 2B 2 V^. ..... (245) 

Lastly, let p have the special value l/4;rr ; then (245) is the 
same as 

potC 2 = A 2 = 2B 2 V^, .... (246) 

which exhibits the divergent vector A 2 in terms of its diver- 
gence B 2 . It is the same as (240), since v# ij/lnr 2 , if r x is 
the nnit vector from B 2 to A 2 . 

The Operation inverse to Slope. 

137. Next, substitute for q in pq a vector, say g. The 
new quantity pg has, being a vector, both curl and divergence^ 
in general. Considering the latter first, we have 

div^?g =p div g + gVp, . . . (247) 

which is an example of (160). Integrating throughout any 
volume, we have 

SNpg = 2divjpg, .... (248) 

as in (159), where N is as before. So, if the surface-integral 
vanishes, we obtain, by (247), 

2^divg= -SgV^, .... (249) 

and, in this transformation, all the vector-analyst has to do is 
to shift the operator v from one operand to the other, and 
change the sign. 

The vector g here has no restriction imposed upon it. It 
may therefore be of the general type = 0! + C 2 , giving 

2^D 2 = - 2 GVp = - 2 G 2 Vp . . . (250) 

Here the portion 2 G l Vp vanishes because Cj is circuital and yp 
is polar, which is one of the important theorems in analysis 
that become visibly true by following the tubes of the circuital 
flux in performing the summation, when the summation is seen 
to vanish separately for every tube. (See 87.) 

If in (250) we give^? the special value l/4?rr, viz., the poten- 
tial due to a unit source at distance r, we obtain 

pot D 2 = B 2 = - 2 CW = - 2 G 2 Vp, . . (251) 


showing how to pass up one step in the divergent series from 
a vector to the preceding scalar. It is the same as (241), 
remembering the value of Vp. 

The Operation inverse to Curl. 

138. Thirdly, we have the curl of pg to consider. Here, 
by (188), 

curing =#> curl g- VgVp, . . . (252) 

Integrating throughout any region, we obtain 

2VN^g = 2curl^g, .... (253) 

which is a case of (162), with pgput for H. So, if the surface- 
summation vanishes, we obtain, by (252), 

2^curlg = 2VgVp, .... (254) 

where the symbol VV is moved from g to p, with a change of 
sign, as before. In this, take > = l/47rr; then, since there is 
no restriction upon g, we get, taking g = C, 

. . (255) 

which is the companion to (251), showing how to pass up one 
step in the circuital series, from O x to B r This is equivalent 
to (229). The divergent part of contributes nothing. That 
is to say, for example, the magnetic force due to a completely 
divergent distribution of electric current, according to Ampere's 
formula for the magnetic force of a current element, is zero. 
We might, indeed, argue from this, that there could not be 
such a kind of electric current ; that is to say, that the current 
must be circuital, since the mathematical machinery itself* 
constructed on old ideas, automatically rejects the want of 
circuitality, and refuses to admit the purely divergent part 
as contributory to magnetic force. This is a perfectly valid 
argument, provided the test of the existence of electric current 
be the existence of magnetic force, which is tantamount to 
what Maxwell insisted upon, in another form. 

For instance, if we calculate by (255) the magnetic force 
due to a supposititious current element at a point, simply by re- 
moving the sign of summation, we obtain the magnetic force of 
a rational current element, a system of circuital current resem- 
bling the induction due to a magnetised particle. (See 62.) 


Remarks on the inverse Operations. 

139. Returning to the three single up-step formulae, it will 
be observed that in every case we base the proof upon the 
vanishing of a surface-integral, viz., 2 Npq, or 2 Npg, or 2 VN>g. 
Now it is obvious that these are true if the surface be taken 
wholly outside the region occupied by the quantity q or g. The 
value pq or pg is made zero all over the surface of summation. 
This is what occurs in the applications made. In (246) the 
space-summation has to include all the B 2 , which repre- 
sents q; it is sufficient, therefore, for the surface to com- 
pletely enclose all B 2 . Again, in (255), the enclosure by the 
surface of all 0, which represents g, will ensure the vanish- 
ing of the surface-integral; and in (251) the same. Now, 
it is usual to imagine the surface to be at infinity, so that 
the space- summations extend over all space. This is also most 
convenient, in general. But caution is sometimes necessary 
when the quantity to be summed extends to infinity. The 
surface-summation then may, or may not, vanish, according to 
circumstances. Even if the quantity summed becomes in- 
finitely small at infinity, the summation may still not vanish. 
To illustrate this, it is sufficient to mention the case of a single 
point-source. The surface-integral of the flux it produces is 
finite always, being the measure of the strength of the source. 
At infinity the flux may be infinitely attenuated, but the sur- 
face is simultaneously infinitely magnified. But in the appli- 
cation of the above processes to practical cases in electro- 
magnetism it is usually quite easy to see that the integral over 
the surface at infinity vanishes, owing either to the actual 
absence of anything to be summed, or else, when there is an 
infinitely attenuated quantity to be summed, of its being, per 
unit area, of lower dimensions than 1/r 2 . The quantity p 
itself attenuates to nothing, as well as q or g, if they are 
existent at all at infinity. 

The three up-step formulae may, by inspection, be transformed 
so as to involve entirely different operations. Thus, in (241), 
put p for 1/471T. Then we have 

B 2 =-pot(C 2 /r) (256) 

Do the same in (240), and we obtain 

C 2 = pot(D 2 /r). (257) 


Finally, (229) gives 

. . . . (258) 

We may sometimes utilise these formulee for purposes of cal- 
culation, should the integrations to be performed be amenable 
to practical treatment. But there is a caution to be men- 
tioned. The quantities whose potentials are calculated by the 
last three formulae are not functions of position, which are de- 
finitely distributed in space, and are independent of the position 
of the point where the potential is reckoned, but are definite only 
when this point is fixed. As you pass to another point, the 
potential there is that of a different distribution from that 
belonging to the first point. These quantities, C 2 /r, &c., are 
therefore not subject to the various properties of the circuital 
and divergent vectors already considered. For example, C 2 is 
purely divergent, but D 2 /r has curl ; or, G 1 is circuital, whilst 
VDj/r has divergence, and so on. 

Integration "by parts." Energy Equivalences in the 
Circuital Series. 

140. The transformations (244), (249), (254), or the more 
general ones containing the omitted surface-summations, ex- 
pressed by putting the sign of summation before every term in 
(242), (247), (252), and converting the summation on the left 
to a surface-summation by the introduction of N, as in (243), 
(248), and (253), are examples of what is, in the Cartesian 
mathematics, called integration " by parts." It is usually a 
very tedious and uninforming process, that is, in Cartesians, 
with its triple and double /'s, its dS and dx dy dz, and its 
I, m, n. The vectorial methods go straight to the mark at 
once, avoid a large amount of quite useless work, and enable 
you to keep your attention fixed upon the actual magnitudes 
concerned, and their essential relations, instead of being dis- 
tracted by a crowd of coordinates and components. 

There are, of course, many other cases which arise of this 
" by parts " integration. One of the most important in con- 
nection with the circuital series of vectors, is the following : 
Substitute for p in the preceding, a vector, say f, then we shall 

2fcurlg = 2gcurlf, .... (259) 

the summations being throughout all space, or, at any rate 


through enough of it to include all the quantities summed. 
The proof is that since 

divVgf=fcurlg-gcurlf, . . . (260) 
which is (178) again, we have, by space-summation, 

2fcurlg = 2gcurlf+2divVgf, . . (261) 

and the third term may be at once turned into the surface-sum- 
mation 2 NVgf. If this vanish, as usual, then (259) follows. 

Applying this result to the series of circuital vectors A 15 A 2 , 
A 3 , &c., each of which is the curl of the preceding, we obtain 
various equivalences. Thus, 

2D 1 2 = 2C 1 E 1 = 2B 1 F 1 = 2A 1 G 1 , . . (262) 

through all space. Operate on one member by curl and on the 
other by (curl)- 1 , to pass from one form to the next. Also 

2 D^ = 2 0^ = 2 6^ = 2 A^, . (263) 

in a similar manner ; and so on. 

In any of these summations we may convert either of the 
circuital vectors involved to a general vector by adding any 
divergent vector. For we see, by (259) that if f is polar, the 
right summation vanishes, and therefore so does the left. So 
2D X 2 is the same as 2DjD, for 20^ = 0. Similarly, 20^, 
is the same as 2 OF lf or the same as 2 1 F, since 2 OjFg = 0, 
and 2 CgFj = 0. It will be understood here that the suffix 1 
refers to the circuital vectors entirely, and the suffix 2 to the 
divergent vectors, and that = 0! + C 2 , &c. 

But we cannot generalise both vectors at the same time. Take 
OF for example. We have, 

2 OF = 2 (Cj + 2 ) (F! + F 2 ) = 2 0^ + 2 2 F 2 , . (264) 

so that the summation of the scalar product of the two diver- 
gent vectors now enters. The series of divergent vectors and 
their intermediate scalars have properties similar to (262), (263). 

Energy and other Equivalences in the Divergent Series. 

141. These properties of the space-integrals of scalar products 
in the divergent series are formally obtainable from the corres- 
ponding ones in the circuital series by changing the suffix from 


1 to 2 ; at the same time changing the type from clarendon to 
roman should the quantity typified be a scalar. Thus, analo- 
gous to (262), (263), we have 

2C 2 2 = 2B 2 D 2 = 2A 2 E 2 =..., . . (265) 
2D 2 2 = 2C 2 E 2 = 2B 2 F 2 = ..., . . (266) 

starting from the square of a vector in the first set, and from 
the square of a scalar in the second. These transformations all 
rest upon (249) ; that is, we pass from any form to dhe next 
by the exchange of v and div between the factors. For 

2 D 2 2 = 2 D 2 div C 2 , by definition of D 2 , 

= - 2 C 2 VD 2 , by integration, using (249), 
= 2 2 E 2 , by definition of E 2 . 

Similarly in all the rest. The preceding equations conveniently 
summarise them. 

But it will be observed that there is another way of pairing 
terms in the divergent series, viz., a vector with a scalar. The 
corresponding transformations do not work so symmetrically as 
the previous. For instance, 

2 C 2 D 2 = - 2 VB 2 .D 2 , by definition of C 2 , 
- 2B 2 VD 2 , by (244), 
= - 2 B 2 E 2 , by definition of E 2 , 
= - 2 div A 2 .E 2 , by definition of B 2 , 
= 2A 2 V.E 2 , by integration. . (267) 

Here the symmetry breaks down. The last transformation 
depends upon 

2(Nf)g = 2(Vf)g = 2(gdivf+f v .g),. . (268) 

which is a case of the theorem (147) or (153). Note that in 
the second form V has to differentiate both f and g, so that the 
full expression is in the third form. If the surface-integral 
vanishes at infinity, we have 

2gdivf= -2fV.g, .... (269) 


which is the transformation used in getting (267). We have 
also, by going the other way, 

2C 2 D 2 = 2C 2 divC 2 =-2C 2 V.C 2 , . . (270) 

by using the same transformation (269). Next take D 2 E 2 . 
Here the vector is the slope of the scalar it is multiplied by, 
whereas in the former case, the divergence of the vector was 
the associated scalar. So it now goes quite differently. Thus, 

2 D 2 E 2 = - 2 D 2 VD 2 = - 2 VJD 2 2 = - 2 N|D 2 2 . (271) 
The result is therefore zero, if the surface-integral at infinity 

The Isotropic Elastic Solid. Relation of Displacement to 
Force through the Potential. 

142. It usually happens that potentials present themselves 
in physical mathematics as auxiliary functions introduced to 
facilitate calculations relating to other quantities. But in the 
theory of the elastic solid, the potential function presents itself 
in a very direct and neat manner; besides that, the elastic 
theory presents excellent illustrations of the above transforma- 
tions and the general theory of v. If the solid be homo- 
geneous and isotropic, there are but two elastic constants, the 
rigidity n and the coefficient of resistance Tc to compression or 
expansion. Let also m = k + Jra, in Thomson and Tait's nota- 
tion ; then the equation of motion is 

f + 7iV 2 G + wVdivG = /oG, . . . (272) 
where G is the displacement, and f the impressed force per 
unit volume, whilst p is the density. Whatever limitations 
may need to be put upon the values of m and n in treating of 
solids as we find them, in speculations relating to ethers they 
may have any values not making the stored energy negative. 
Thus m may, as we shall see presently, go down to the nega- 
tive value - n, keeping n positive. 

Split f into f x + f 2 , and G into G x + G 2 , where f x and G T are 
circuital, and the others divergent. Then (272) may be split 
into two equations, one circuital, the other divergent. Thus, 
remembering (193), 

pGy . . . (273) 

/oG 2 . . . . (274) 


In the circuital equation Jc does not appear, so that the propa- 
gation of circuital disturbances depends only upon the rigidity 
and density ; speed, (n/p)l. In the divergent equation both n 
and k appear, and the speed of disturbances of this class is 
{(n + m)/p}l. In another form, if we take the curl of (272), the 
divergence goes out, so that the curl of G, which is twice the 
rotation, has the curl of f for source, and is propagated inde- 
pendently of compressibility. 

And if we take the divergence of (272), we see that the 
divergence of G, or the expansion, has the divergence of f for 
source, and the rate of its propagation depends upon both 
rigidity and compressibility. But when m = 0, there is but one 
speed of propagation, and a complete amalgamation of the two 
kinds of disturbances. 

In equilibrium, when equilibrium is possible, the right 
members of the last three equations vanish, since they repre- 
sent "rate of acceleration of momentum" (a long-winded 
expression). We therefore have 

. . . (276) 

The solutions are visible by inspection. We see that nGr l is 
the potential of f lt and (n + m)G 2 the potential of f 2 . That is, 
the displacement produced by circuital impressed force is its 
potential divided by n ; and the displacement produced by 
divergent impressed force is its potential divided by n + m ; 
whilst, in the general case, we must split the impressed force 
into circuital and divergent parts, and then re-unite them in 
different proportions to obtain the resultant displacement. 




In the case of incompressibility Jc, and therefore m, is infinite, 
so that G 2 is zero, and the displacement is pot fjn, whatever 
f 2 may be. For f 2 is balanced by difference of pressure, which 
is set up instantly. The corresponding speed is infinite, but 
there is no displacement. 


If, again, m = 0, and there is but one speed, the displacement 
is the potential (divided by n) of the impressed force, whatever 
be its type. 

On the other hand, if ra + w = 0, we see by (274) that there 
is no steady state possible due to divergent force, as f 2 is then 
employed simply in accelerating momentum on the spot. The 
corresponding speed is zero. The circuital disturbances are 
propagated as before. This is the case of Sir W. Thomson's 
contractile ether, in which the wave of normal disturbance is 
abolished by making the speed zero. 

The Stored Energy and the Stress in the Elastic Solid. The 

Forceless and Torqueless Stress. 

143. We may also find expressions for the stored energy 
from the equation of motion. The work done and stored by 
the impressed force f is JfG, though, if f be put on suddenly, 
an equal amount is dissipated, since, G being the final displace- 
ment, fG is the work then done by f, per unit volume. Now 

by (264). Calling the first part U^ and the second part Ug, 
we have, by (273), (274), 

U 1= -2|wG 1 V 2 G 1 = 2Jn(curlG)2 .... (280) 
U 2 = - 2 i(m + w)G 2 V 2 G 2 = 2 \(m + n) (div G) 2 . (281) 

The transformations here used are (259) for Uj and (249) for 
U 2 . We see that the energy stored is expressed in terms of 
the squares of the rotation and of the expansion. In the 
contractile ether U 2 is zero, and the stored energy is U r 
We do not correctly localise the energy by the above formulae, 
but only express the total amounts. For correct localisation 
we need to know the stress and the distortion. Now the 
distortion is a function of the variation of displacement, so is 
known in terms of G. Can we, however, find the stress itself 
from the equation of motion ? If not, we can come very close 
to it. Thus, in a state of equilibrium, F + f=0, if F is the 
force arising from the stress. So, by (275), (276), 

..... (282) 
and F =/tV 2 G + wV 2 G 2 ..... (284) 


Now let P N be the stress on the plane whose normal is N. 
We have, by consideration of the equilibrium of a unit cube, 

FN = divP N , (285) 

to express the relation between an irrotational stress and the 
force arising from it. Therefore, applying this to (284), 

divP N = 7iV 2 GN + wV 2 G 2 N. . . . (286) 

Here the divergence of P N is given ; find P N itself. The 
immediate answer is, by (201), or (214), 

PN= -VpotdivP N , 
or P N = ttVGN + mVG 2 N, . . . (287) 

provided P N is divergent. That is, we have constructed a 
stress-vector giving the proper force required. But, without 
interfering with this essential property, we might add on to 
the right side of (287) any circuital vector. That we must do 
so now, we may see by remembering that the stress must be 
irrotational, or produce no torque, and then by finding that 
(287) does give a torque. To show this, consider the first 
part only of the stress (287), say with m = 0. Then 

NP M - MP N = 7i(NV.MG - MV.NG). 
Here take M=j and N = k; then 

kPj - jP k = tt(V 3 G 2 - V 2 G 3 ) = - ni curl G, 

by (149). This gives the i component of the torque, which is 
therefore - n curl G. But G 2 has no curl, therefore the second 
part of (287) produces no torque. We require, therefore, to 
add to P N in (287) a stress giving no force, but a torque n curl G. 
Such a stress is 

X N = ncurlVGN (288) 

as may be tested in the above manner. 
This brings us from (287) to 

P N = W (VGN + curl VGN) + mVG 2 N, . . (289) 

which is a stress-vector giving the correct force and no torque. 
[It should be noted here that since (285) applies to an irro- 
tational stress, the process employed is only justified when we 
finally get rid of the torque, as in (289), (290). If the stress is 


rotational, the divergence of P N is really the N component of 
the force due to the conjugate stress. Thus the force due to 
nv.GN is nVdivG, and the torque ncurlGr. The force due 
to ?ftV.G 2 N is mvdivG, with no torque. The force due to 
n curl VGN is ncur! 2 G, and the torque n curl G. Adding 
together the three stresses and the three forces we obtain the 
stress (289), with no torque and the correct force.] 

But on comparison with the real stress deduced from the 
elastic properties of a solid, which is 

P N = tt(V-GN + NV.G) + N(m-rc)divG . . (290) 

we see that they do not agree. Yet they can only differ by 
a stress which produces neither force nor torque. And we 
know already, by (288), that if G = G 2 , the modified stress will 
give no force or torque. In fact, on comparing (289), (290), 
we find that their difference is of this nature, (290) being 
equivalent to 

P N = rc(VGN + curl VGN) + m(VG 2 N - curl VG.N), (291) 

where the first and third terms on the right are sufficient to 
give the correct force, but with a torque, which is, however, 
cancelled by the second term ; whilst the fourth term is 
apparently (so far as force and torque are concerned) like a 
fifth wheel to the coach, off the ground. If we inquire under 
what circumstances the real stress can assume this singular 
form, we shall find that m + n = and curl G = will do it. 
With these conditions, only the circuital part of P N is now left, 
and (291) reduces to 

P N = 2curlVG 2 N (292) 

It is the case of irrotational displacement in the contractile 
ether, previously referred to, and is entirely remote from the 
real elastic solid. The noteworthy thing is that we cannot 
apparently conclude what the stress is, even when the force 
and torque to correspond are everywhere given, owing to the 
forceless and torqueless stress coming into the formulae. The 
form (292) is convenient for showing at sight that there is no 
force. It may, by (181), remembering the constancy of N, be 
expanded to 

P N = 2w(NV.G 2 -NdivG 2 ), , , . (293) 


where put N = i, j, k in turns to get the three stresses on the 
planes having these normals. One-third of the sum of the 
normal tractions on these planes is - (4^/3) div G 2 , or, which is 
the same, + Jc div G. It is the negative of the pressure. But 
in the real stress (291) itself, the fourth part, from its having 
the negative sign prefixed, correctly associates pressure and 
compression. But this forceless and torqueless stress does not 
contribute anything to the total energy. The amount of 
stored energy is 

= - J2 (F^ + F 2 G 2 + F 3 G 3 ) 

div P 2 + G s div p s) 
P 3 VG 3 ), .... (294) 

where the first line needs no remark, the second is got by 
(285), and the third by the common transformation (249), P p 
etc., being the stresses on the i, j, k planes. The final form in 
(294) shows the correct distribution of the energy in terms of 
the stress and the distortion. Now, if in the stress P N be in- 
cluded any terms giving rise to no force, the above transforma- 
tion shows that they contribute nothing on the whole to the 
energy of distortion. This is the case with the fourth term in 
(293), and would also be the case with the second term, only 
that we have no right to ignore it, on account of the torque 
thereby brought in. If the solid be not unbounded, it is 
sufficient for its boundary to be at rest for the same principles 
to apply. 

[The practical meaning is that in the contractile ether the 
energy of distortion due to any irrotational displacement is 
zero on the whole, the sum of the positive amounts in certain 
parts being equal to the sum of the negative amounts in the 

Other Forms for the Displacement in terms of the Applied 


144. The simple solution (279) may, of course, receive 
many other forms. If it be desired to find the displacement 
due to an explicitly given impressed forcive, it is a matter of 
some importance to select a method of obtaining it which shall 
not be unnecessarily difficult in execution j for different pro- 


cesses leading to the same result may vary greatly in readiness 
of application. Now, if the impressed forcive be either circuital 
or divergent, we do not need to modify. But if of a mixed 
type, then it may be desirable. Put t l = t- f 2 , then an alter- 
native form is 

* & J* ^ 

; . . , (295) 

and now, when f is given, it is only the part pot f 2 that needs 
development. We have 

. . (296) 
as before explained. Now, if we notice that 

W = 2/r, ..... (297) 

which may be proved by differentiating, we can further con- 
clude that 

. . (298) 

a very curious result. The potential of a radial vector following 
the inverse-square law of intensity is a radial vector with a 
constant tensor in all space. Using this result, we have 

.. (299) 
so that (295) becomes 

. . (300) 


wherein f alone appears on the right side. 
Another form is got by using 

f 2 = V2fV ? 7, /. potf 2 = Vpot2fVp. . (301) 
But a better one is 

potf 2 = 2/v^/87r, .... (302) 

where s is length measured along f, and / is the tensor of the 
latter. This makes 

O-Ipotf 5/?^Z/v* . . . (303) 

n n 

which is generally suitable for practical calculation. 



A considerably more complex form is given in Thomson and 
Tait. It may be obtained from (303) by means of the identity 

where s l is unit s, and therefore parallel to f, making 


ds r 

the same as Thomson and Tait's formulae when expanded in 
cartesians. But this is a gratuitous complication, as (303) is 
simpler in expression and in application. Of course (300) is 
simpler still in expression, and the practical choice may lie 
between it and (303), or (295). 

Notice that an impressed forcive of the divergent type with 
a single point-source produces not merely uniform radial dis- 
placement, as per (298), but an infinite discontinuity, in fact, a 
disruption, at the source itself. It is a very extreme case, of 

The Elastic Solid generalised to include Elastic, Dissipative, 
and Inertial Resistance to Translation, Rotation, Ex- 
pansion, and Distortion. 

145. The elastic solid with two elastic constants (k and n) 
has not been found sufficiently elastic to supply a thoroughly 
satisfactory analogy with Maxwell's ether, though partial 
analogies may readily be found. Other kinds of elasticity 
than resistance to compression and distortion, and other kinds 
of resistance than elastic resistance, present themselves to the 
consideration of searchers for analogies between the propa- 
gation of disturbances in Maxwell's ether and in the brutally 
simple elastic solid of theory, which is, however, known to 
fairly represent real solids within a certain range. 

There are four distinct ideas involved in the displacement 
of a small portion of matter, viz., the translation as a whole, 
the rotation as a whole, the change of size, and the change of 
shape. These separate themselves from one another naturally. 
As regards the mere translational motion, if it be only iner- 
tially resisted, we have the equation of motion 



where p is the density, q. the velocity, f impressed force (per 
unit volume), F tfae force arising from the stress associated 
with the strain (including rotation, distortion, and expansion), 
and 8/St the time-differentiator for moving matter. This equa- 
tion is constructed on simple Newtonian principles. We may, 
however, wish to have elastic and frictional resistance to trans- 
lation, as well as inertial resistance. Then generalize the 
above to 

f +F- ( ft + ft * +*)<, . . . (307) 

where G is the displacement, when small departures from 
equilibrium are concerned. Here p 2 is the p of (306), p l is 
the frictionality (Lord Kelvin's word for coefficient of friction), 
and p Q is an elastic constant. The displacement from equi- 
librium calls into action a back force p G proportional to the 
displacement, with storage of potential energy ; a force /a 1 G- 
proportional to the velocity, with waste of energy ; and a force 
/o 2 G proportional to the acceleration, with associated kinetic 
energy. The potential energy is J/o G 2 , the rate of waste ftG 2 , 
and the kinetic energy Jp 2 G 2 . 

As regards the force F, this is given by (284), when the stress 
is irrotational, and the elastic constants are n and k. But, in 
general, we may proceed thus. First separate the rotation 
from the strain vector. We have 

NV.G = J(NV.O- + V.NG) + J(NV.G - V.NG) 

= p N -JVNcurlG (308) 

Here Ny.G means the variation of displacement per unit dis- 
tance along N, which is any unit vector, so that by giving N 
all directions (practically only three) the complete state of 
strain is known. This strain vector is above analysed into the 
strain p N without rotation, and the second part depending upon 
rotation. But p N includes the expansion, as well as the dis- 
tortion, or mere change of shape. To exhibit the distortion 
without expansion, one-third of the expansion (vectorised) must 
be deducted. Thus 

NV.G = (p N - IN div G) + iN div G - JVN curl G (309) 



shows the separation of the strain into a distortion (without 
change of size), an expansion, and a rotation, which are 
naturally independent, 

If distortion, expansion, and rotation are all elastically 
resisted, three independent elastic constants (in an isotropic 
medium) intervene between the above strain and the corre- 
sponding stress P N upon the plane whose normal is N. To 
obtain P N , multiply the distortional part by 2n, the expansional 
part by 3&, and the rotational part by 2v, and add the results. 

PN = 2/t(p N - JN div G) + Nfc div G - vVN curl G. (310) 

Compare with (290). Here n and k are as before, whilst v is a 
new elastic constant connected with the rotation, and which 
was previously assumed to have the value zero. That is, there 
was assumed to be no resistance to rotation. The corres- 
ponding torque is 

S = 2vcurlG ...... (311) 

With the rotational part of the stress is also associated trans- 
lational force, given by 

- curl JS= -v curl 2 G. . . . (312) 

Adding this on to the former expression for the force (or 
deriving the force from (310) directly), we find that 

F = w(V 2 G + JV div G) + &V div G - v curl 2 G (313) 

represents the translational force to be used in the equation of 
motion (307). We should notice here that the quantity 
k div G in (310) represents a uniform tension. It equals one- 
third of the sum of the normal tractions on any three mutually 
perpendicular planes. Its negative represents the pressure, or 
p = - k div G. Now when there is incompressibility, k is 
infinite, and div G is zero. But their product usually remains 
finite. So in any case we may replace the k term in (310) by 
- Np, and the k term in (313) by - Vp. 
The energy of the strain is 

. . (314) 

or one-half the sum of the scalar products of the stress vector 
and the strain vector for three perpendicular planes, P v P 2 , P 3 


being the stresses on the i, j, k planes respectively. On 
reckoning up, by (310), (309), we find the energies of expan- 
sion, rotation, and distortion are all independent, and that U 
is their sum, given by 

U = n[ Pl 2 + p 2 2 + p 3 2 _ l(div G)2] + J&(div G)2 + 1 v(curl G)', (315) 

where the v term is the energy of rotation ( = \ torque x 
rotation), the Jc term is the energy of expansion, and the n 
term the energy of distortion. 

Now the stress (310) is derived from the strain (309) by 
the introduction of elastic resistances only. There is, however, 
no reason why we should limit the nature of the resistance in 
this way. Consider the n term only for example, relating to 
distortion. There may also be dissipative or frictional resist- 
ance to distortion. To exhibit it, change n to n Q + n^d/dt) in 
(310), (313). Then n Q will be the rigidity and n the viscosity, 
or coefficient of frictional resistance to distortion. For instance, 
if we abolish n and v, and retain n v and k, we have the stress 
in a real viscous fluid according to Stokes's theory. It may be 
that there is not a complete disappearance of the rigidity in 
a fluid ; if so, then retain both n and n v When the rigidity 
is marked, as in a solid, n Q is the important part of n ; whilst 
in a fluid it is the other part. It is, however, not at all to be 
expected that the expression of the viscosity of solids by n v if 
true at all, would extend beyond the small range of approxi- 
mately perfect elasticity. Lord Kelvin's experiments on the 
subsidence of the oscillations of wires tended to show that a 
different law was followed than that corresponding to the 
viscosity of fluids, and that elastic fatigue was concerned in 
the matter. 

There may also be, conceivably, inertial resistance to dis- 
tortion, and not only elastic, but also frictional and inertial 
resistance to rotation, and the same with respect to expansion. 
These will be brought in by generalising the three elastic 
constants k, n, and v of (310) thus : 

. . (316) 


where of the nine coefficients on the right side; the first three 
are elastic constants, the second three dissipative constants, the 
last three inertial constants ; the elastic constants involving 
potential energy, the dissipative constants- waste of energy, and 
the inertial constants kinetic energy, similarly to (307) as 
regards translation. Out of these twelve constants only four 
are in use, or seven, if v and v,, and p-^ be included, which 
additional have been speculatively employed. This does not 
represent finality by any means ; but sufficient has been said 
on the subject of generalising the elastic constants to emphasize 
the fact that there is plenty of scope for investigation in the 
theory of the motion of media of unknown internal constitution, 
which, externally viewed, involve the four elements translation, 
rotation, expansion and distortion. 

The activity of the stress vector P N is P N q, and the flux of 
energy is (see 68, 72 also), if q be the speed, or tensor of the 

Qtf, ..... (317) 

where U and T are the complete potential and kinetic energies, 
per unit volume, and Q N is the stress conjugate to P N , to 
obtain which change the sign of the last (rotational) term in 
(310). The convergence of the energy flux represents the 
work done and stored or wasted in the unit volume. Consider- 
ing the term - Q g ^ only (the rest being the convective flux 
of energy), its convergence is 

div (Q l2l + Q 2 ? 2 + Q 3?3 ) = VjtoQ) + V 2 (P 2 q) + V 3 (P 3 q) 

- & div Qi + & div Q 2 + q 3 div Q 3 + Q x Vft + Q 2 V? 2 + Q 3 V<? 3 

. . (318) 

where P is the force as in (313). Fq. is the. translational 
activity, which may, by (307), be employed in increasing kinetic 
and potential energy or be wasted. The rest represents the 
rate of increase of the sum of the three potential energies when 
there is merely elastic resistance but if we use (316) we 
obtain also the sum of the rates of waste of energy, through 
&P n v v v and also the rate of increase of the sum of the kinetic 
energies, through & 2 , n 2 , v 2 . 


Thus we have, by (310), 
1 V 1 + P 2 V 2 + P s V 3 )q 

= ^ ( 2n(p, - Ji div G) + M di v G - vVi curl G ) 
dx\ / 

- Jj div G) + Jcj div G - vVj curl G 

+ 2(p s - Jk div G) + k div G - vVk curl G 
dz\ / 

= 2w(p, VjCt + P 2 V 2 q + p 3 V 3 q) - ^ div G div a 

+ & div G div q + v curl G curl q . . . (319) 

Take for illustration, the rotational term v curl G curl q. This 
is simply (dldt) {}>v (curl G) 2 }, or the rate of increase of the 
rotational energy, when v is a constant, but when we use the 
last of (316) it becomes 

) + v^curl a) + ^(i" 2 (rl a)*), (320) 

showing the rates of increase of potential and kinetic energy, 
and the rate of waste (the middle term) due to rotational 
friction. Similarly as regards the other terms in (319). The 
expansion terms give 

), . (321) 

where the middle term is the rate of waste, and the others 
rates of increase of potential and kinetic energies. Finally, 
the distortional terms of (319) give 

(p, 2 + pV + p' 3 2 - 


where the first line is the rate of increase of the potential 
energy, the second line the frictional rate of waste, and the 
third line the rate of increase of kinetic energy. We thus 
complete the energy relations of the stress with the generalised 


elastic constants, and so far as small motions are concerned, 
the equations are manageable. 

Electromagnetic and Elastic Solid Comparisons. First 
Example : Magnetic Force compared with Velocity in 
an incompressible Solid with Distortional Elasticity. 

146. Comparisons between the propagation of electromag- 
netic disturbances in Maxwell's ether, or in a homogeneous 
isotropic dielectric, whether conducting or not, and the propa- 
gation of motion in an elastic solid, either simple or generalised, 
may be made in a variety of ways. They all break down 
sooner or later, but are nevertheless useful as far as they go. 
A few cases will be now considered, based on the preceding. 
First take the case of a non-conducting dielectric at rest, and 
compare it with the regular elastic solid made incompressible 
by infinite resistance to compression. The incompressibility 
includes inexpansibility. 

In the dielectric we have, if p stands for d/dt, for convenience, 

curl(H-h) = cpE, curl (e - E) = /*pH, . (323) 

the circuital equations connecting E and H ( 38, 24, 66), 
where e and h are impressed, and //., c are the inductivity and 
permittivity. Now, the equation of motion in an incompressible 
solid is, by (273), 

. . (324) 

where p and n are the density and rigidity, G the displacement, 
q. the velocity, and f x the circuital part of the impressed forcive, 
the divergent part being inoperative. Or we may take the 
impressed forcive to be circuital to begin with. 

Now, let h = 0, and e be finite. If it has no curl, it does 
nothing, as before shown ( 89). The source of disturbance is 
curl e. This being the case in the dielectric, and f x being the 
source of motion in the solid, it will be convenient to take f x 
and curl e as corresponding. Let the latter be (temporarily) f. 
Then (323) gives (remembering that div H = 0), 

f = upB. + curl E = /^H + curl 2 H/cp, 

f=w - H = ^ 2 -- ' ' (325) 


Comparing this with the second of (324), and assuming the 
equality of f and f 1? we see that H is velocity, /* density, c~ l 
rigidity ; and so on. IL/p means the time-integral of H. Or, 
more explicitly, let Z be the time-integral of H, so that H=>Z; 

f=(^2-c-iV 2 )Z (326) 

which compares with the first of (324). Thus 

Z = H/p corresponds to G, (spacial displ.) 
(mag. force) H = pZ q, (velocity), 

(el. current) curl H curl q, (2 x spin), 

(el. displ.) D = cE curl G, (2 x rotation), 

(inductivity) /* p, (density), 

(permittivity) c ,, l/n, (compliancy), 

(mag. source) f= curie f p (impd. force), 

(mag. energy) J//H 2 |pq 2 , (kin. energy). 

This is, as far as it goes, an excellent analogy, particularly on 
account of its directness and ease of application, and on account 
of the similarity of sources. The disturbances of H generated 
by f in the dielectric and propagated away at speed (/*c)~* are 
precisely represented by the velocity generated by similarly 
distributed impressed Newtonian force, which is propagated in 
the same manner through the solid at the equivalent speed 
(n/p)%. Of course the correspondence at similar moments also 
applies to the other quantities compared. 

But it is imperative (in general) that the motions in the 
solid should be small. The disturbances should therefore be 
of the fluctuating or alternating character, so that Z, the 
time-integral of H, does not mount up. For, correspondingly, 
if this were allowed, then G would mount up, and the elastic 
solid get too much out of shape to allow us to treat d/dt at a 
fixed point and 8/8t for a moving particle as identical. 

Observe also, that the electric energy -JcE 2 is not matched 
by the potential energy of distortion ; although their total 
amounts are equal, their distributions are entirely different. 
This is a fatal failure in detail. Nor have we any right to 
expect that a scheme like Maxwell's, depending upon rotations, 
can be perfectly matched by one which depends on shears. 


Second Example: Same as last, but Electric Force 
compared with Velocity. 

147. Another analogy of the same class is got by making 
the displacement in the solid represent the time-integral of E 
in the dielectric. Thus let e = G in (323), and - curl h = + g. 
It is now g that is the source. Eliminate H between the two 
equations (323). We get 

epE + g = curl (- curl E)//*p. . . . (327) 
Or, since divE = (when there is no electrification) 

g-(q,-2!Wg*-2_V . . (328) 

\ ftp/ \ [A/ 

if A = E/p, making E = pA. 

Comparing (327) with (324) for the solid, we see that we 
have E representing velocity, ft compliancy, c density, and so 
on, being a general turning over of all the quantities. Thus 

A = E/p corresponds to G (displacement), 

c p, 

(el. force) E q, (velocity), 

H racurlG, 

curl h f x (circuital impd. force). 

The conclusions are similar as regards the generation and 
propagation of disturbances of the two kinds compared. There 
is a similar failure to before as regards the energy of dis- 
tortion. It ought to be the magnetic energy J/*H 2 now, but 
it is not. This want of correspondence as regards one of the 
energies will be removed (and can only be removed) by doing 
away with the rigidity and substituting something else, which 
must, however, be equivalent in its results as regards the 
propagation of disturbances from place to place. 

Third Example : A Conducting Dielectric compared with a 

Viscous Solid. Failure. 

148. As we see from the preceding that the propagation 
of electromagnetic disturbances through a non-conductor can 


be imitated in an incompressible solid, possessing the usual dis- 
tortional rigidity, let us next introduce electric conductivity 
on the one hand, and examine what changes are needed in the 
analogies of 146, 147 to keep them working, if it be possible. 
Since there is waste of energy associated with the conduction 
current, it is clear that some sort of frictional force requires to 
be introduced into the elastic solid. 

First try distortional friction. The equation of motion of 
an incompressible viscous solid is 

fi-^ 2 -(o + Mv 2 ]Gi ' (329) 

where f x is the impressed force per unit volume, G the displace- 
ment, p the density, n Q the rigidity, and n^ the associated fric- 
tionality. Also, p stands for d/dt, for practical convenience in 
the operations, whether direct or inverse. Thus, p~\ or qjp 
means the time-integral of q, which would in the ordinary 

notation of the integral calculus be expressed by / qcft, a nota- 

J o 

tion which is not convenient for manipulative purposes in in- 
vestigations of this class. We may derive (329) from (324) by 
changing n to n^ + n^ ; or by (306), (313). [See also p. 229.] 

Now the circuital equations of E and H in a conducting 
dielectric (homogeneous, isotropic, stationary), are 

curl (H - h) = (k + cp) E, curl(e-E)=/xpH, . (330) 

comparing which with (323) we see that the effect of introducing 
conductivity is to change cp to k + cp, so that the equation of 
H is got by making this change in (325), giving 

< < 331 > 

where Z = H/p. But on comparing this equation with (329), 
we see that the viscosity in the one case and the conductivity 
in the other enter into the equations in different ways. We 
can only produce a proper correspondence when the coefficients 
of y 2 are equal; or, in another form, when the operators 
p(k + cp)~ ! and (n Q + n } p) are equivalent in effect. This is not 
generally possible. But there are two extreme cases of agree- 
ment. One we know already, viz., when there is no conduc- 


tivity and no viscosity, as in 146. The other is when there is 
no permittivity and no rigidity, which is sufficiently important 
to be separately considered. 

Fourth Example: A Pure Conductor compared with a 
Viscous Liquid. Useful Analogy. 

149. This is the extreme case of a pure conductor, or a 
conductor which cannot support elastic displacement. It dissi- 
pates energy, but does not store it electrically ; though, on the 
other hand, its magnetic storage capacity is retained. It 
is compared with an incompressible viscous solid with the 
rigidity abolished; that is to say, with an incompressible 
viscous liquid. Put c = in (331), and we have the equation 

f=(/^-ArV)H; . . (332) 

and putting n Q = in (329), with q the velocity substituted for 
p&, gives us the equation 


in the liquid. These admit of immediate comparison. Observe, 
however, the curious fact that whereas in 146 the permit- 
tivity was the reciprocal of the rigidity, so that the vanishing 
of one means the infinitude of the other, yet now both the 
permittivity and the rigidity vanish together, as if they were 
equivalent. This emphasises the incompatibility of (329), (331). 
Our present analogy must stand by itself, all idea of permit- 
tivity and rigidity being thrown away. 

We compare magnetic force H in a pure conductor with the 
velocity q in a viscous liquid ; the inductivity ft with the 
density of the liquid, so that the magnetic energy and the 
kinetic energy are compared, and the current is twice the 
spin. Also the source f = curie in the case of the conductor is 
compared with circuital impressed force in the liquid. So far is 
the same as in 146. But now, in addition, we have the electric 
resistivity k~ l represented by the liquid viscosity. We conclude 
that the disturbances q generated by f,_ in the liquid are similar 
to the disturbances H in the conductor generated by f, and 
that the propagation of electrical disturbances in a conductor 
is like that of motion iii a viscous incompressible liquid. It 


takes place by the process called diffusion. It is the limiting 
case of elastic wave-propagation, with distortion and dissipation 
of energy by friction. 

We are usually limited to small motions in this analogy, so 
that the impressed forces should not act in one sense continu- 
ously, but should fluctuate about the mean value zero, for the 
reason mentioned before. But we are not always limited to 
small motions. In certain symmetrical distributions of mag- 
netic force and current we may remove this restriction 
altogether. In the liquid case (333) p strictly means 8/8t, 
whilst in (332) it means d/dt. In the former a moving particle 
is followed ; in the latter we keep to jane place. But in the 
cases of laminar flow referred to, S/St reduces to d/dt readily 
enough, so that the analogy is carried on to cases of steadily 
acting impressed forces. 

The analogy is an important one, owing to the readiness with 
which the setting in motion of water by sliding friction can be 
followed, as when a wind blows over its surface. Two interest- 
ing cases are the penetration of magnetic induction and (with 
it) electric current into a core enveloped by a solenoidal coil, 
in whose circuit an impressed force, variable or steady, acts ; and 
the penetration of magnetic induction and electric current into a 
straight wire when an impressed force acts in its circuit. Consider- 
ing the latter we may state the analogy thus. Understanding, 
in the first place, that the circumstances should be such that 
the influence of electric displacement outside the wire is negli- 
gible, we may replace the wire by a similar tube of water, and 
then replace the impressed voltage in the electric circuit by a 
uniform tangential drag upon the surface of the water in the 
direction of the length of the pipe. The current of water 
in the pipe and the electric current in the wire, will 
vary similarly under the action of similarly varying impressed 
forces. Thus, a steadily applied force on the water will first 
pull the outermost layer into motion; this, by the viscosity, 
will pull the next layer, and so on up to the axis. The initial 
current is purely superficial ; a little later we have a central 
core in which the water is practically motionless, surrounded 
by a tubular portion whose outermost layer is moving 
rapidly, and innermost very slowly ; later still, the whole mass 
is in motion, though less rapidly at the axis than at the boundary; 


finally, the whole mass of water acquires a state of uniform 
motion. Substituting electric current for current of water, we 
obtain a representation of the way the electric current is set 
up in a wire, passing through the various stages from the initial 
surface current to the final uniformly distributed current. If 
the impressed force be rapidly oscillatory, we stop the penetra- 
tion as above described in its early stage, so that if the frequency 
be great enough, there is a practical confinement of the current 
(in sensible amount) to the skin of the wire (or of the water 
respectively). But if the mean value of the oscillatory force 
be not zero, it is the same as having two impressed forces, one 
steady, the other periodic, without any bias one way or the 
other. Then we have finally a steady current throughout the 
wire, plus the oscillatory current with superficial concentration. 
When a straight core is magnetised in a solenoid, it is the 
magnetic induction which is longitudinal, or parallel to the axis. 
Comparing this with the current of water in the pipe, we have 
the same state of things as before described, as regards the 
penetration of magnetic induction into the core. 

One effect is to increase the resistance of a wire when the 
frequency is great enough. It will be remembered that Prof. 
Hughes (in 1886) brought forward evidence of this increase, 
and, therefore, in the opinion of others, of the truth of the 
theory of surface conduction along wires under certain circum- 
stances which was advanced by me a year previous. There has 
since been plenty of confirmatory evidence of a more complete 
nature ; that is, not merely of an approximation towards, but 
of an almost complete attainment of surface-conduction. 

This action of conductors is sometimes referred to as magnetic 
screening. It should, however, be noted that the screening is 
done by the conductor itself, not by the currents " induced" in 
the outer part of the wire or core, which are, as it were, merely 
a sign that the screening is taking place. The screening action 
often seems to be merely superficial ; but this is accidental, 
from the disturbances being communicated to the conductor 
from without. In reality, any and every part of a conductor is 
screened from the rest by the portion immediately surrounding it, 
that is, by its skin, and by that only. It is primarily the con- 
ductivity that causes the screening, so that it is rather conduc- 
tive screening than magnetic screening. The result is that dis- 


turbances can only pass through a conductor by the (relatively) 
very slow process of diffusion, so unlike that by which they are 
transferred through a non-conductor. But the property of con- 
ductivity acts conjointly with the indue tivity, so that although 
copper would be the best screener on account of its high conduc- 
tivity, the great inductivity of iron usually far more than com- 
pensates for its inferior conductivity, and makes iron take the 
first place as a screener. 

Fifth Example : Modification of the Second and Fourth. 

150. As we were not able to make an analogy with c and 
k both finite, likewise n and n lt when H was taken to be 
velocity, let us try the effect of taking E to be velocity, as in 
147. Change the source from f to g= -curlh; then we 
have merely to alter cp in (328) to k + cp to obtain the required 
modification. Thus, 


if A = E/p. Here, again, on comparison with (329), we see 
that distortional friction will not furnish a proper analogy in 
the general case. There seem to be but two special cases 
possible; first, with the conductivity and frictionality both 
zero (not the resistivity and frictionality), which reproduces 
the case in 147 ; and, secondly, the permittivity and the 
rigidity both zero (not permittivity and compliancy), which 
brings us to the viscous liquid again. Thus, putting c = in 
(334) brings us to 


which compares with (333). We now compare conductivity 
with density, and inductivity with the reciprocal of the 
viscosity. This is by no means so useful as the previous form 
of the viscous fluid analogy, although it is fundamentally 
of the same nature, involving propagation by diffusion. 

As in 146, 147, we had a failure of representation of one 
of the energies by the energy of distortion, so now, in the 
viscous fluid analogies, we fail to represent the dissipation ol 
energy properly. The latter depends, as shown before, on the 
velocity of distortion, so we might expect a similar failure. 


Sixth Example : A Conducting Dielectric compared with an 
Elastic Solid with Translational Friction. 

151. But we can get a working analogy to suit the con- 
ducting dielectric in another way. Thus, still keeping the 
distortional elasticity or rigidity, give up the viscosity or dis- 
tortional friction, and substitute translational friction. For 
the equation of motion of an incompressible solid with the usual 
constants, rigidity n and density /o 2 , and with translational 
frictionality p 1 is 

t^(p lP + P^-n^)Q,. . . . (336) 

where f a is circuital impressed force. Now operate on (331) 
by 1 + k/cp, producing the equation 

- < 337 > 

Comparing this with (336) we see that there is a proper cor- 
respondence of form on the right sides, though not on the left. 
The circuital impressed force fj_ is replaced, not by f, but by 
f + M/cp. This is awkward for the analogy as regards the gene- 
ration of disturbances. But away from the sources, there is a 
useful correspondence. As in 146, Z corresponds to spacial 
displacement in the solid, H to velocity, ju- to density, c to com- 
pliancy (distortional) ; whilst now, in addition, the translational 
frictionality in the solid has the complex representative pk/c in 
the dielectric ; or &(/xr) 2 , if v is the speed of propagation of 
disturbances. So disturbances of H are propagated in the same 
way through a conducting dielectric, as are disturbances in an 
incompressible solid with the usual rigidity, with frictional 
resistance to displacement superadded. But we do not cor- 
rectly localise the energy dissipated. The rate of waste per 
unit volume is &E 2 in the dielectric, whilst it is /^q 2 in the solid, 
which corresponds to &(/xv) 2 H 2 in the dielectric. This being 
entirely against Joule's law renders this form of analogy 
unsatisfactory. But we may easily make a substantial improve- 
ment by taking E to be velocity in the solid. 

Seventh Example : Improvement of the Sixth. 
152. Thus, let g= - curl h be the source of disturbance in 
the dielectric, so that we have the equations (334) to consider. 


Take the second form and compare it with (336), and we see at 
once that there is a considerably more simple correspondence 
than in the case last treated. Thus, the source f x of circuital 
displacement G- corresponds to the source g of circuital A (the 
time-integral of E) ; the electric force now means velocity ; /A 
is compliancy ; c is density ; k itself is the frictionality ; the 
electric energy JcE 2 is kinetic energy ; and the waste (Joulean) 
is properly localised in the solid by the frictional waste of energy 
Pi<l 2 per second. The induction ^H is twice the rotation. In 
fact, this case is the same as in 147, with translational fric- 
tion added without destroying the analogy. It is clear that the 
correct localisation of the waste, and the correspondence of 
sources, makes the analogy be more readily followed. Of course, 
there is the same failure as regards the distortional energy as 
in 147. 

Eighth Example: A Dielectric with Duplex Conductivity 
compared with an Elastic Solid with Translational 
Elasticity and Friction. The singular Distortionless 

153. As a last example in which use is made of distortional 
elasticity, let us generalise the non-conducting dielectric by the 
introduction of magnetic conductivity. We have 

curl(H-h) = (& + cp)E, . . . (338) 
curl (e-E) = for + flp)H, . . . (339) 

where g is the magnetic conductivity. So, if curl h = 0, and 
we eliminate E, we have 

. . (340) 

Or, if curl e = and we eliminate H, we have 

. . (341) 

We see that we cannot, when k and g are both finite, get a 
satisfactory analogy as regards the generation of disturbances 
by f p circuital impressed force in a solid. Consider, therefore, 
only the free propagation, subject to the equation 

, . . (342) 


where for Z we may substitute H or E or other electrical quan- 
tities. Since, in (342), we have a term proportional to Z itself 
(or its substitute), we must introduce elastic resistance to dis- 
placement in the elastic solid to make an analogy. Thus, 

. . (343) 

is the equation of motion in the incompressible solid if there be 
elastic and frictional as well as inertial resistance to translation, 
by (307), (313), 145. 

Now expand (342) and divide by c. We get 

which is suitable for comparison with (343). When Z is the 
spacial displacement, H is velocity, c compliancy, /* density, as 
in 146 ; but in other respects the correspondence is complex, 
for the elastic resistance to translation p Q is kg/c, whilst the 
frictionality p l is g + kp/c. 

If we take E to be velocity there will be another set of corre- 
spondences, c being density and p compliancy. 

However unsatisfactory the analogy maybe in details, we have 
still the result that disturbances in a dielectric with duplex 
conductivity (electric and magnetic) are propagated similarly to 
motions in an elastic solid constrained in the manner above de- 
cribed. The effect of the two kinds of distortion and waste of 
energy involved in the existence of the two conductivities are 
therefore imitated by one kind of frictionality in the elastic solid, 
assisted by elastic resistance to translation. 

The singular distortionless case ccmes about when 

4/W>2 = />i 2 in the solid, 

k/c=g/{j, in the dielectric, 

In either case the effect of the frictional resistance on the dis- 
turbances is made merely attenuative. Disregarding the atten- 
uation with the time, the propagation takes place in the di- 
electric in the same way as if it were non-conducting, and in the 
solid in the same way as if it were devoid of frictional and 
elastic resistance to translation. In the electromagnetic case 
we may follow this into detail with ease in a symmetrical 
manner. It is, however, rather troublesome in the elastic 

} ^ 
j ' 


solid, on account of the want of correspondence in detail, due 
to the employment of distortional elasticity, and for other 

The Rotational Ether, Compressible or Incompressible. 

153 A. All the above analogies, however good or bad they 
may be in other respects, are deficient in one vital property, 
inasmuch as they involve the elasticity of distortion, and with 
it, the energy of distortion. But the electromagnetic equations, 
especially when put in the duplex form symmetrical with 
respect to the electric and magnetic sides, have nothing in 
them suggestive of distortional forces, nor can we represent 
either the electric or the magnetic energy as the energy of 
distortion. On the other hand, the equations are fully 
suggestive of rotation. If, then, the elastic solid has still 
to do duty for purposes of analogy, and yet not fail upon the 
point mentioned, it is clear that the rigidity must be done 
away with, at least as an electromagnetically active influence. 
Its place must then be taken by elastic resistance to rotation, 
by using a rotational instead of a shearing stress. This 
brings us to the medium invented by Lord Kelvin, called by 
him simply Ether, and contrasted with Jelly, which means 
an incompressible elastic solid with ordinary rigidity. This 
rotational ether has mass of course, for one thing, which 
brings in kinetic energy (of translation) ; and, on the other 
hand, it possesses, by means of internal arrangements with 
which we are not immediately concerned, the property of 
elastically resisting rotation, and consequently stores up 
energy of the potential kind, to balance the kinetic. If this 
were all, there would be little more to say. But Lord Kelvin 
found that his Ether enabled him to complete a long-delayed 
work, viz., to produce a satisfactory elastic-solid analogy to 
suit the problem of magnetic induction (" Mathematical and 
?hysical Papers," Vol. I., art. 27, and Vol. III., art. 49) ; and 
I >ave pointed out that this rotational ether enables us to 
construct good all-round analogies for the propagation of dis- 
turbances in a 'stationary dielectric (Appendix to chap. II., 
ante), with correct localisation of both energies and of the 
flux of energy. As these are the only all-round elastic-solid 
analogies yet known, we may here consider them again, 


especially in relation to the previously given analogies, and 
to show where the changes come in. 

Go back to equation (313), giving the translational force P 
due to the stress in an elastic solid when there is rotational 
elasticity as well as rigidity. We have 

F = n (V 2 G + JV div G) + JfcV div G - v curl 2 G, (346) 

where G is the displacement, n the rigidity, k the compressive 
resistivity, and v the new elastic constant connected with the 
rotation. Now, we have curl 2 = V div - V 2 , so we may rearrange 
(346) thus, 

F = ( + v)V 2 G + (& + Jtt-v)VdivG, . (347) 

and this may be split up into circuital and divergent parts, viz., 

(circuital), . (348) 
2 , (divergent). (349) 

Now observe that in the circuital equation, n and v occur 
additively, so that equal parts of each have the same effect, 
and either may be replaced by an equal amount of the other. 
This is irrespective of the compressibility, which is not con- 
cerned in circuital disturbances. We see, then, that the 
rigidity may be done away with altogether and equal v substi- 
tuted, without affecting the propagation of circuital disturb- 
ances. This is a very remarkable property. 

Observe, also, that in the divergent equation v does not 
enter at all, although n does, and in the same way as in the 
ordinary compressible rigid solid. In fact, the coefficient k + ^n 
is the same as the former m + n, of 142 and later. So the 
propagation of divergent disturbances is the same as if the 
rotational elasticity were done away with. 

Now, we have already noticed the case in which the speed 
of the divergent wave is brought down to zero the contractile 
ether of 142, 143. In an ordinary rigid solid, which was 
then referred to, this requires negative compressibility. But 
now, if n be abolished and v substituted, so that the circuital 
propagation is unaffected, the divergent equation will contain 
only k t so that to make the speed of the divergent wave zero 
we need only abolish the resistance to compression, or make 
& = 0. The medium does not now resist either change or 


shape or of size, but resists rotation only, and the potential 
energy is the energy of rotation. This is one extreme form of 
the rotational ether, made by combining it with the contractile, 
by the evanescence of the rigidity and of the resistance to com- 
pression. The medium is quite neutral as regards expansion 
and compression. 

Now, it is clear, from equation (348) and matter in previous 
paragraphs, that we may construct analogies to suit the non- 
conducting dielectric by abolishing n and using v instead, 
without concerning ourselves with k at all, which may have any 
value from zero to infinity. But Lord Kelvin's ether is got by 
going to the other extreme from the neutral case just men- 

First Rotational Analogy : Magnetic Force compared with 


154. Take & = oo (with n = 0) making the medium incom- 
pressible, and therefore making disturbances in an unbounded 
medium be necessarily circuital, whether the impressed forcive 
be circuital or not. We now have 

...... (350) 

by (348), whilst the other equation (349) will merely assert 
that F 2 is balanced by difference of pressure. This we are not 
concerned with, since our impressed forcive should be circuital 
in the electromagnetic comparisons. The stress (310) is now 

P N = -vVNcurlG, .... (351) 

where we ignore the uniform pressure for the reason mentioned. 
The torque accompanying this stress is, by (311), 

S = 2vcurlG, ..... (352) 
and the translational force is 

F= -curljS=i>v 2 G, (353) 
agreeing with (350). So the equation of motion is, by (306), 

fi = (^ 2 'vV2)G = (^-^V 2 )q, . . (354) 

where fj is the circuital impressed force, and q. is the velocity. 
As before, we keep to small motions in general. 


Equation (354) may be at once compared with the electro- 

f=(^-V 2 \ H = (^_Vf\ z> . . (355) 
\ cpJ \ c/ 

as in (325), (326), our first example, and we therefore deduce 
the following correspondences : 

Z = H/p stands for G, (spacial displ.) 

(mag. force) H ,, q, (velocity), 

(el. current) curl H curl q, (2 x spin), 

(el. displ. ) D ,, curlG, (2 x rotation), 

(inductivity) /x, ,, ,, /a, (density), 

(permittivity) c ,, ,, 1/v, (rotl. compliancy), 

(induction) B ,, /oq, (momentum), 

(el. force) E ,, JS, (J x torque), 

(mag. energy) J/xE 2 /oq 2 , (kin. energy), 

(el. energy) JcE 2 ,, Jv(curlG) 2 , (rotl. energy), 

(source of H) f= curie ,, f lt (impd. force), 

(energy flux) VEH VJSq, (energy flux) 

Referring to the similar list in 146, it will be seen that the 
correspondences there given are repeated here, except that the 
elastic constant changes its meaning, and that there are now 
additional correspondences. The electric energy is correctly 
localised by the potential energy of the rotation. Since the 
kinetic energy is also correctly localised, we need not be 
surprised to find that the energy-flux has the same distribution. 
The activity of the stress P N is P N q per unit area, which is 
the same as g-NP^, if P q be the stress on the plane whose 
normal is q, provided the stress is irrotational. But if ro- 
tational, as at present, it is ^NQ q , where Q is conjugate to P. 
This activity means energy transferred from the side of the 
plane when the stress vector is reckoned, to the. other side, or 
against the motion. The vector expressing the energy flux is 
therefore -^Q q , not counting the convective flux. In our 
present case, by (351), Q is the negative of P, because it is 
purely a rotational stress, so W = qP q is the vector flux of 
energy ; or, by (351), 

W - - vqV^ curl G - - vVq curl G, . (356) 


which is, by the above table, the same as VEH, the electro- 
magnetic flux of energy. (In the above, q and ^ are the tensor 
and unit vector of q.) 

The above reasoning is applied directly to the unit element 
of volume. But it may be easier to follow by taking am- 
volume into consideration. If N is the normal outwards from 
the surface enclosing it, then P N is the pull, per unit area, of 
the matter outside, on the matter inside the surface, across the 
unit area of the interface ; and P N q. is its activity. The total 
activity of P N all over the surface is therefore 2P N q, which is 
the same as - 2 N^P^, and expresses the work done per second 
by the matter outside on the matter inside the surface, or the 
rate of transfer of energy from the outside to the inside ; and 
its equivalent is the rate of increase of the stored energy, 
potential and kinetic, within the surface. The convergence of 
the vector qP^ expresses the same property for the unit volume, 
so that <?P q is the flux of energy (per unit area) itself. (The 
small convective flux of energy is ignored here.) 

Circuital Indeterminateness of the Flux of Energy in general. 

155. That any circuital flux of energy may be siiperadded, 
without making any difference in the transformations of 
energy, is a fact which is of importance as evidence against the 
objectivity of energy, but is of no moment whatever in the 
practical use of the idea of a flux of energy for purposes of 
reasoning. We should only introduce an auxiliary circuital 
flux when some useful purpose is served thereby. 

I may here remark that (speaking from memory) when Pro- 
fessor J. J. Thomson first objected to the VEH formula as 
representative of energy-flux, by reason of the circuital inde- 
terminateness, he added that a knowledge of the real flux 
could only be determined from an actual knowledge of the 
real dynamical connections involved, that is to say, by a know- 
ledge of the mechanism. But this is surely an inconclusive 
argument, because the circuital indeterminateness applies even 
then. In fact, it is universal in its application, as energy is at 
present understood. For there to be an absolutely definite flux 
of energy through space seems to require energy to have objec- 
tivity in the same sense as matter, which is a very difficult notion 
to grasp, and still more difficult to accept. But even if it be ac- 


cepted, the argument of circuital indeterminateness remains in 
action, when it is desired to find what is the flux in a given case. 

In a recent number of the Phil. Mag., Mr. Macaulay, by the 
addition of a certain circuital flux to VEH, brings out the 
result that in stationary states the modified flux reduces to 
pG, where p is potential and current-density. I am, how- 
ever (unless I forget much of what I have learnt in the last 
15 years), unable to see that the auxiliary flux proposed serves 
any useful purpose. In the first place, it greatly complicates 
the flux of energy in general, and is entirely against the more 
simple ideas to which we are naturally led in the study of 
electromagnetic waves, whether in dielectrics or Conductors. 
Besides that, it is implied that the function p, or electric 
potential, is a determinate quantity specifying some definite 
state of the medium. This seems to me to be an idea which 
has no place at all in Maxwell's theory. It may have place in 
other theories, but that is not to the point. To exemplify, 
put a closed circuit (with battery) supporting a steady current 
inside a metal box, and electrify the latter from outside. 
According to the pG formula, the flux of energy in the wire 
suffers a very remarkable change, by reason of the raising of 
the potential of the box and its interior. But, according to the 
interpretation of Maxwell's scheme which I expound, there is 
no change whatever in the electrical state of the interior of the 
box in the steady state (though there may and must be a 
transient disturbance in the act of charging it, which is a 
separate question), and the energy distribution and its flux are 
the same as before, because E and H, which settle the state of 
affairs, electric and magnetic, are unchanged. There are, 
however, I believe, some electricians who will be much gratified 
with the pG flux in steady states. 

The total flux of energy through a wire will be pC, where by 
C we mean the total current, obtained from pG by integration 
over the cross-section. Now this is somewhat suggestive of 
my expression VC for the total flux of energy along a circuit of 
two conductors. But there are radical differences. For VC 
is the integral form of VEH in the dielectric. Although C in 
VC means the same as in pC, being the circuitation of H, or 
the gaussage, V has no connection with p the potential, for it 
(V) is the line integral of E across the circuit from on& wire 


to the other i.e., the transverse voltage. Again, the VC 
formula holds good in variable as well as in steady states, 
whilst the proposed pG holds for steady states only. Even in 
steady states, when V specializes itself and becomes difference 
of potential, it remains different from p. 

But although I cannot see the utility of the proposed change, 
which seems to be a retrograde step, I think that Mr. 
Macaulay's mathematics, which is of a strong kind, may be of 
value in the electromagnetic field in other ways, especially 
when cleared of useless and treacherous potentials. 

Second Rotational Analogy : Induction Compared with 

156. Returning to the suppositional Ether, the conception 
is of an incompressible medium possessing mass, which involves 
translational inertia, and therefore kinetic energy, and which 
elastically resists rotation, and so stores potential energy ; and 
we have supposed that the velocity of the medium means 
magnetic force, and that its density means /x, so that the 
kinetic energy and the magnetic energy are compared. But 
we may equally well have this comparison of the energies com- 
bined with a different interpretation of /A. Take, for instance, 
the induction to mean velocity. Then, since the magnetic energy 
is J/^B 2 , we see that it is now, not /x, but fjr 1 that is the density, 
whilst H is momentum. Also, since /xcv 2 = 1 = pv~ l v 2 , we see 
that the new interpretation of c is vy~ 4 . That is, the change of 
[j, from density to its reciprocal involves the change of c from 
compliancy to its reciprocal multiplied by v~^ making p 2 /v or p/v' 1 , 
But it is better to write them out side by side, thus : 

(mag. energy) J/*~ 1 B 2 stands for 

(mag. force) 
(el. current) 
(el. displ.) 
(el. force) 
(el. energy) 




(kin. energy), 

p curl q, 
p curl G, 

vp~ l curl G, 

(potl. energy). 


The potential energy still correctly localising the electric 
energy, in spite of the other changes, we may expect the flux 
of energy to be correct. We have 

VEH = vp~ l V curl G . />q = - vVq. curl G, 

as before, equation (356). 

Similarly, we may assume that p is not the density, nor its 
reciprocal, but an unstated function of the density, say /A =f(p), 
and work out the various correspondences that this necessitates. 
We shall, for example, by the formulae for magnetic and kinetic 
energy, require 

Of course, in thus comparing magnetic with kinetic energy, we 
shall always have H compared with q, or with some multiple 
of q, which may change its " dimensions," as, for example, in 
the above detailed cases, where H is changed from velocity to 

Probability of the Kinetic Nature of Magnetic Energy. 

157. If it be asked why, in the previous analogies, a pre- 
ference has usually been shown for the representation of magnetic 
force (or a constant multiple thereof) by velocity, the answer 
would be, substantially, that it has been done in order to make 
the magnetic energy be kinetic. Now, it is true that in a per- 
fectly abstract electromagnetic scheme, arranged in duplex 
form in which every electric magnitude has its magnetic re- 
presentative and, therefore, including a magnetic conduction- 
current with waste of energy, there would be a perfect 
balance of evidence as regards the kinetic nature of eiiher 
the electric or the magnetic energy, when the other is to be 
potential energy, as of a state of strain in an elastic medium. 
For, if it were argued from a certain set of relations that 
the magnetic energy was kinetic, the force of the argument 
could be at once destroyed by picking out an analogous set of 
relations tending to show (in the same way as before) that 
the electric energy was kinetic. 

But, as matters actually stand, with an imperfect corre- 
spondence between the electric and magnetic sides of the elec- 
tromagnetic scheme, there seems to be a considerable pre- 
ponderance of evidence in favour of the kinetic nature of the 


magnetic energy. We may refer, in particular, to the laws of 
linear electric circuits, which were shown by Maxwell to be 
simply deducible by the ordinary equations of motion (gene- 
ralised) of a dynamical system, on the assumption that 
magnetic energy is kinetic; there being one degree of 
freedom for every circuit, and the variables being such 
that every linear electric current is a (generalised) velocity. 
In this theory, the dielectric in which the conducting circuits 
are immersed is regarded as unyielding, so that electric dis- 
placement cannot occur in it, and the currents are confined 
entirely to the conductors. 

Now, we could construct a precisely similar theory of con- 
ductive magnetic circuits immersed in a medium permitting 
displacement, but destitute of magnetic inductivity. The 
circuital flux of magnetic induction in the former case would 
now be replaced by a similar circuital flux of electric dis- 
placement; the former electric currents becoming magnetic 
currents, and the magnetic energy becoming electric energy. 
But this electric energy, when expressed in terms of the linear 
magnetic currents, would possess the property of allowing us 
to deduce from it the laws of the magnetic circuits, by using 
the generalised equations of motion, on the assumption that 
electric energy is kinetic, in a manner resembling Maxwell's 
deduction of the laws of electric circuits. Therefore, sup- 
posing the state of things mentioned to really exist, we might 
become impressed with the idea that electric energy is kinetic ; 
just as, at present, it seems hardly possible to avoid entertain- 
ing the idea of the kinetic nature of the magnetic energy. 

We do not, however, need to go to generalised dynamics to 
arrive at this probable conclusion. It is sufficient to start 
with a sound general knowledge of dynamical facts and prin- 
ciples, not necessarily mathematical, but such as may be ac- 
quired in practical experience by an intelligent and thoughtful 
mind, involving clear ideas about inertia, momentum, force, 
and work, and how they are practically connected (the Act of 
Parliament notwithstanding). On then proceeding to the expe- 
rimental study of electrokinetics, including the phenomena of 
self-induction in particular, the dynamical ideas will be found to 
come in quite naturally. Lastly, the generalised theoretical 
dynamics will serve to clinch the matter. Although adding 


little that is novel, it will corroborate former conclusions, and 
co-ordinate the facts in a compact and systematic manner, suit- 
able to a dynamical science. 

On the other hand, there is little that is suggestive of 
kinetic ideas in electrostatics, whilst there is much that is 
suggestive of the potential energy of a strained state. This 
fact, combined with the kinetic suggestiveness of the facts (and 
the equations embodying them) in which magnetic induction 
and electric currents are concerned, explains why the associa- 
tion of magnetic with kinetic, and electric with potential 
energy becomes natural. It is, therefore, somewhat a matter 
of surprise, as well as rather vexing, to find that in order to 
extend the last-considered rotational analogy to a conducting 
dielectric, we must, if we wish to do it simply, give up the com- 
parison of magnetic force with velocity and electric force with 
torque, and adopt the converse system, making the electric 
energy be kinetic, and the magnetic energy the potential 
energy of the rotation. 

Unintelligibility of the Rotational Analogue for a Conduc- 
ting Dielectric when Magnetic Energy is Kinetic. 

158. We can, perhaps, most easily see that this plan should be 
adopted by writing down the two circuital equations of electro- 
magnetism, and then, immediately under them, the correspond- 
ing proposed circuital equations of the rotational ether. Thus, 
taking H to be velocity, as before, we have, to express the first 
circuital law in the dielectric and its mechanical companion, 

. . . (358) 

Here the terms in vertical are to be compared. Magnetic 
force H becoming velocity q, the electric force E becomes |S, 
or half the torque ; c the permittivity is the compliancy v~ l , 
and gj is a newly-introduced companion to g, which is the source 
- curl h. It will be interpreted later. 

The other circuital equations are 

.... (359) 
. . (360) 


We now have the additional analogues of inductivity and 
density, and f, which is the source curl e, is compared with t v 
which is impressed translational force per unit volume. 

So far relating to a non-conducting dielectric, and giving an 
intelligible dynamical analogy, if we wish to extend it to a con- 
ducting dielectric, according to Maxwell's scheme (including, of 
course, a pure conductor which has, or is assumed to have, no 
permittivity), we require to change cp in the first circuital law 
(357) to k + cp, whilst the second circuital law needs no change. 
But the right member of (358) needs to be changed to match 
the modified (357). Thus, 

g + curl H = (Jc + cp)E, .... (361) 
g 1 + curU = ft + v-^)(iS), . . (362) 

where ^ is the new coefficient, to match k. But what is its 
interpretation in the rotational ether, and how is the latter to 
be modified to make & x as intelligible as the other constants ? 

Now, since we use rotational elasticity to obtain the po- 
tential energy, and we associate the rotation with the electric 
displacement, it is suggested that rotational friction should be 
introduced to cause the waste of energy analogous to that of 
Joule. But if there were frictional resistance proportional to 
the spin, we should have 

S = 2(v + Vjjp)ourlG, .... (363) 

instead of (352), connecting the torque -with the rotation. 
But this will not harmonise at all with (362). We cannot do 
what we want by rotational friction, but require some special 
arrangement, whose nature does not appear, in order to interpret 
(362) intelligibly. 

The Rotational Analogy, with Electric Energy Kinetic, 
extended to a Conducting Dielectric by means of Trans- 
lational Friction. 

159. But, by changing the form of the analogy, choosing 
the electric energy to be kinetic, the extension to a conducting 
dielectric can be made in a sufficiently obvious manner. Put 
(360) under (361), thus 

g + curl H = (k + cp)E, .... (364) 
. . . (365) 



OH. in. 

where we Introduce p l to match k. 
to get the other pair, thus 

Also put (358) under (359) 


We have now a fit, with an intelligible meaning to be given to 
the new coefficient that brings in waste of energy. For (365), 
which is compared with the first circuital law, is the trans- 
lational equation of motion in the rotational ether when there 
is frictional resistance to translation, expressed by p 1 q, so that 
P 1 is the frictionality, and /a^ 2 the rate of waste. 
The correspondences in detail are as follows : 

(el. force) 
(el. displ.) 
(el. energy) 
(mag. force) 
(mag. energy) 
(cond. current) 
(true current) 

stands for G, 
,< Q, 




(spacial displ.). 




(kin. energy). 

( - | torque). 



Jv(curlG-) 2 , (rotl. energy). 
Pa ' (frictionality). 

(rate of waste). 

These should be studied in connection with the circuital 
equations (364), (366), and their matches underneath them, if 
it be desired to obtain " an intelligent comprehension " of the 
true nature of the analogy, and to correct any errors that may 
have crept in. 

If we keep away from the sources of energy, there is little 
difficulty in understanding the analogy in a broad manner. 
But the sources require special attention before the electro- 
magnetic and rotational analogues are intelligible. We know 
that activity is the product of two factors, a " force " and a 
" velocity." Newton knew that. In modern dynamics, too, 


where the velocity is not a primitive velocity, but has a 
generalised meaning the time-rate of change of some variable 
the corresponding force generalised (not a primitive force) is 
still such that the product of " force " and " velocity " is 
activity, or activity per unit volume, &c., according to con- 
venience. In our electromagnetic equations, for instance, EC 
and HB and ED are activities (per unit volume), and we call 
E and H the " forces " (intensities), and the other factors the 
fluxes, the corresponding " velocities." Similarly, eO, eD, and 
hB are activities (per unit volume understood), where e and li 
are intrinsic, communicating energy to the system. 

Now, since the " variables " may be variously chosen in a 
dynamical system, we need not be surprised if it should some- 
times happen that the (generalised) force turns out to be a 
(primitive) velocity, and the (generalised) velocity to be a (primi- 
tive) force. Here is food for the scoffer, for one thing. At 
any rate, we should be careful not to confound distinct ideas, 
and remember the meaning of the activity product. Our pre- 
sent rotational analogy furnishes an illustration. We have the 
equation of activity, 

. (368) 
where W = V(E - e) (H - h). . . (369) 

W is the energy-flux, Q the rate of waste, U the electric, and 
T the magnetic energy per unit volume, whilst the left member 
represents the rate of supply of energy by the intrinsic sources 
e and h, being the sum of their activities. 

Now what are the analogues of e and h ? Remember that 
only their curls appear in the circuital equations, and that f and 
g are the sources of disturbances. By inspection of (365), 
(367), we see that f x is the curl of a torque, and g l the curl 
of a velocity, say 

2f x = curl S , g l = - curl q . . . (370) 

Then, just as the analogue of H is - JS, the analogue of h is 
-|S . Impressed magnetic force is, therefore, represented 
by an impressed torque per unit volume, and its activity is 

(-4S )v-X-iS), . . . (371) 


which is, of course, plain enough. But the other activity is 

Qo (ft + /*>), ( 372 ) 

where q is the analogue of e. Here the supposed " force " has 
become a velocity, and the " velocity " a force. For the factor 
of q in (372) is the analogue of electric current- density, and 
means the (primitive) force per unit volume in the rotational 
ether, partly employed in increasing momentum, partly work- 
ing against friction. Of course this perversion is rather 

The rotational analogues of (368), (369) may be readily 
written down by proper translations in accordance with the above, 
remembering (370). 

Mr. W. Williams, who has recently (Physical Society, 1892) 
published a very close study of the theory of the " dimensions " 
of physical magnitudes, on applying his views to the electro- 
magnetic equations and the rotational ether, has arrived at the 
conclusion that the representation of either E or H by velocity 
leads to the only two systems that are dynamically intelligible. 
It should be remembered, however, that his conclusion is 
subject to certain limitations (mentioned in his paper) regarding 
dimensions, for otherwise the conclusion might seem to be of 
too absolute a nature, as if one of E or H must be velocity. In 
any case, I cannot go further myself at present than regard the 
rotational ether as furnishing a good analogy, which may lead 
later to something better and more comprehensive. The pre- 
sent limitation to small motions (in general) is a serious one, 
and there are other difficulties. 

Symmetrical Linear Operators, direct and inverse, referred 
to the principal Axes. 

160. Linear vector functions of a vector play a very 
important part in vector-algebra and analysis, just as simple 
equations do in common algebra. Thus, in the theory of 
elasticity the strain vector is a linear function of the direction 
vector, and the stress vector is another linear function of the 
direction vector. In electromagnetism the electric displace- 
ment is a linear function of the electric force, and so is the 
conduction current-density ; whilst the magnetic induction is a 
linear function of the magnetic force, when its range is small 


enough. There are, therefore, at least five examples of linear 
vector functions to be considered in electromagnetism. In 
other sciences too, the linear function often turns up, and 
frequently in pure geometry, when treated algebraically. 
Finite rotations may be treated by the method, and in the 
algebra of surfaces of the second order the linear connection 
between two vectors is prominent. It would be inexcusable 
not to give some account of linear operators in this chapter 
on vector-analysis and its application to electromagnetism. 
The subject, however, is such a large one that it is only 
possible to deal with its more elementary parts in a somewhat 
brief manner not, however, so brief as to be useless. 

The simplest kind of linear connection is that of Ohm's law 
in isotropic conductors. We have C = E, where E is the 
intensity of electric force and C the current-density, whilst k 
is a constant, the conductivity. No specification of direction 
is here made. Bub when vectorised, we have the equation 
C = E instead, Tc being as before. We now assert that E 
and are parallel, whilst their tensors are in a constant ratio, 
for all directions in space. 

Suppose, however, we alter the conductivity of the body in 
a certain direction, say that of i (e.g., by means of compression 
parallel to i), making it k v whilst its conductivity is k 2 in all 
directions transverse to i. We now have Cj_ = /^E^ and also 
O x = &}}, when the electric force is parallel to i. But if it be 
transverse to i, then we have C = & 2 E. There is, in both cases, 
parallelism of E and 0, but the ratio of the tensors changes 
from &L to Jc 2 . What, then, is the current when E is neither 
parallel nor transverse to i ? The answer is to be obtained by 
decomposing E into two components, one parallel to i, the other 
transverse ; then reckoning the currents to match by the above, 
and finally combining them by addition. Thus 

C = i. 1 E 1 +j. 2 E 2 + k.& 2 E 3 , ... (1) 
where E I} E 2 , E 3 are the i, j, k scalar components of E. We 
have here made use of the linear principle. It is that the sum 
of the currents due to any two electric forces is the current 
due to the sum of the electric forces. This principle (suitably 
expressed) applies in all cases of linear connection between two 
vectors, and is the ultimate source of the simplicity of treat- 
ment that arises. 


More generally, let the conductivity be different in the 
three co-perpendicular directions of i, j, k, being k lt k y k z re- 
spectively. Then we have 

= *E-U 1 E 1 +J.fc 8 E 2 + kJfc 8 E s . ... (2) 

The current is now coincident with the electric force in three 
directions only, namely, along the principal axes of conductivity. 
When E has any other direction than that of a principal axis, 
we have no longer parallelism of and E, and their relation 
is expressed by equation (2). It is the general type of all 
linear relations of the symmetrical kind when referred to the 
principal axes, if we remove the restriction which obtains in 
the electrical example that the &'s must be all positive. Given, 
then, that = JcE, with the understanding that is a symme- 
trical linear function of E, so that k must represent the linear 
operator connecting them (the conductivity operator in the 
above example) the full answer to the question, What is the 
corresponding to a given E? is obtainable in the above 
manner, viz., by first finding the axes of parallelism of E and 
and the values of the principal &'s, and next, by adding together 
the three C's belonging to the three component E's parallel to 
the axes. 

The inverse question, Given E, find 0, is similarly answerable. 
For if C = />E, where p is the operator inverse to k (or the 
resistivity operator in the case of conduction-current), we know 
that ^ft = 1, if p l is the constant resistivity parallel to i, and 
similarly k 2 p 2 = 1 and & 3 /o 3 = 1 for the j and k axes, so that in 

... (3) 

where C v C 2 , C 3 are the scalar components of current. 

Precisely similar remarks apply to the permittivity operator 
c in D = cE, connecting the displacement D with the electric 
force E, and to the inductivity operator /A in B = /*H, connect- 
ing the induction B with the magnetic force H, when the con- 
nection is a linear one. It is not without importance to con- 
stantly bear in mind the above process of passing from E to C 
through &, and the process of inverting k to /o, because when 
treated in a general manner, with reference to any axes, the 
relations become much more complicated, whilst their intrinsic 
and resultant meaning is identically the same. 


Geometrical Illustrations. The Sphere and Ellipsoid. 
Inverse Perpendiculars and Maccullagh's Theorem. 

161. Next let us obtain some geometrical illustrations of 
the pre tdous. The transition from to is like that from a 
sphere to an ellipsoid. If a solid be uniformly subjected to 
what is termed a homogeneous strain, any initially spherical 
portion of it becomes an ellipsoid. If this be done without 
rotation, the state is such that all lines parallel to a certain 
axis, say that of i, in the unstrained solid, are lengthened or 
shortened in a certain ratio in passing to the strained state, 
without change of direction. The same is true as regards lines 
parallel to two other axes, say j and k, perpendicular to each 
other and to the first axis, with different values given to the 
ratio of lengthening or shortening in the three directions of 
preservation of parallelism. 

The equation of the ellipsoidal surface itself is 

when referred to the centre and the principal axes. Here a, 6, c, 
are the lengths of the principal semi-axes, whilst x, y, z, are 
the scalar components of r, the radius vector from the centre 
to any point of the surface. It may be written 

= N 2 say, . . (5) 

from which we see that when r belongs to the ellipsoid, N is 
a unit vector, the vector radius of a certain sphere from which 
the ellipsoid r may be obtained by homogeneous strain, the 
three ratios of elongation being a, &, c (that is, the co-ordinate 
x/a in the sphere becomes x in the ellipsoid, and so on). 
Now, if we square (3) and divide by E 2 , we may write it 

where the C's are the components of 0, and E is the tensor of 
E. Now suppose E is constant, and let E take all directions 
in succession. Its extremity will range over a spherical 
surface, whilst the end of the corresponding or &E will 
range over an ellipsoidal surface. For we may take ^E, # 2 E, 
fe 3 E in (6) to be a, b, c in (4) ; when G I} C 2 , C 3 will simulta- 
neously be x, y, z. The semi-axes of this current-ellipsoid are 



the principal currents in the directions of parallelism of electric 
force and current. 

Similarly to (6), we have, by squaring (2), 

( 7 ) 

from which we see that when C is constant, so that the vector 
ranges over a spherical surface, the corresponding E simulta- 
neously ranges over the surface of an ellipsoid whose principal 
semi-axes are the principal E's. 

There are other ways of illustration than the above. Thus 
we may see from the form of the right member of (4) that it is 
the scalar product of r and another vector s, thus 

rs = l, ....... (8) 

where r and s are given by 

..... (9) 

Here we see that s is a linear function of r, and such that if 
N = <r, then s = <N = <<r = < 2 r. See equation (5). The in- 
terpretation of the new vector s is easily to be found. It is the 
reciprocal of the vector perpendicular from the centre upon the 
tangent plane to the ellipsoid at the extremity of r. For if p 
be this perpendicular, the projection of r upon p is evidently 
p itself, because the three vectors r, p, and the line in the 
tangent plane joining their extremities form a right-angled 
triangle whose longest side is r. Therefore 

rp=P 2 , ....... (11) 

or, dividing by p 2 , 

rp- 1 -!, ....... (12) 

comparing which with (8), which is true for all r's of the 
ellipsoid, we see that s = p" 1 , as stated 
Now we have 

= -l=^il+JV- + JL (13) 

EC ftEC /> 2 EC /o 3 EC' 

from which we see that if we construct the ellipsoid whose 
principal semi-axes are (p 1 EO) i , and so on, the radius vector 
will be E and the corresponding reciprocal perpendicular will 
be C/EO. Here EC is understood to be constant. 


Similarly, we have 

i 2 2 2 

so that in the ellipsoid with principal semi-axes (^EC)*, &c., the 
radius vector is C and the reciprocal perpendicular is E/EC. In 
both these methods of representation neither E nor has con- 
stant tensor, but their product, the activity, is constant. 

In the case (6), where it is E that is considered constant, the 
reciprocal of the perpendicular on the tangent plane to the 
ellipsoid of is 

. _ ii . = 
* E 2 E 2 ' 

This is also an ellipsoid. For E is an ellipsoid, and pE only 
differs in the principal constants being the reciprocals of those 
in the former case. 

Similarly, in the case (7), where C is constant, the reciprocal 
perpendicular is 

= c 
C 2 ' 

and again the extremity of JcG ranges over an ellipsoidal 

Thus the sphere of E with constant tensor is associated with an 
ellipsoid &E, whose reciprocal perpendicular is />E/E 2 or ^r~ 1 E/E 2 , 
another ellipsoid. It follows that the reciprocal perpendicular 
of the latter gives us the former ellipsoid again. 

This reciprocal relation of two ellipsoids through their in- 
verse perpendiculars is an example of Maccullagh's extraor- 
dinary theorem of reciprocal surfaces. It may be stated thus 

1^ = 1, and r 2 s 2 = l, .... (17) 

be the equations of two surfaces referred to the same origin, 
T l and r 2 being the radius vectors, and s 1} S 2 the reciprocals of 
the vector perpendiculars from the origin on their tangent 
planes. If, then, we choose r 2 to be B I} we shall simultaneously 
make S 2 be r r That is, starting with the surface r lt construct 
the surface whose radius vector r 2 is B V the reciprocal perpen- 
dicular of the ij surface. Then the reciprocal perpendicular s 2 
of the second surface (s 1 or r 2 ) will be the radius vector r x of 
the first surface. 


This admits of simple vectorial proof. For if ijSj = 1 be the 
equation of a surface according to the above notation, we 
obtain, by differentiation, 

r^Sj + s^ = 0, (18) 

if di l and efSj are simultaneous variations of r x and s r But di l 
is in the tangent plane to r 15 and therefore at right angles to 
BH so that s^ij = 0. This leaves T l ds l = also. But ds 1 or 
c?r 2 equivalently is in the tangent plane of the surface whose 
vector radius is S 1} or r 2 ; therefore r x is parallel to the perpen- 
dicular on the same, or parallel to S 2 , say, ij = xs 2 , from which 
r 1 s 1 = a;s 1 s 2 or #r 2 s 2 . But 1^ = 1 and r 2 s 2 = l, so ==!, and 
ij = s 2 , which completes the connections. 

In terms of the perpendiculars, let p l and p 2 be their lengths 
and r v r 2 the corresponding radius vectors, then 

*i'- r*=Pi'- Pv ( 19 ) 

and P! is parallel to p 2 , and r 2 to p r 

Internal Structure of Linear Operators. Manipulation of 
several when Principal Axes are Parallel. 

162. Leaving now the geometrical illustrations connected 
with the ellipsoid, consider the symmetrical linear operator by 
itself. If D = cE, where c is the linear operator, we know by 
the above exactly how it operates through the principal o's. 
We can, however, write c in such a manner that it shall state 
explicitly its meaning. Thus, if c v c 2 , c 3 be the principal c's 
belonging to the axes of i, j, k, we have 

cE = i.c 1 E 1 +j.c 2 E 2 + k.c 3 E 3 , . . . (20) 
by 160. Now put the E's in terms of E, producing 

cE = i.c 1 iE+j.c 2 jE + k.c 3 kE. . . . (21) 

In this form, the operand may be separated from .the operator, 

cE = (i.c 1 i+j.c 2 j+k.c a k)E, . , . (22) 

so that the expression for c itself is 

c = i.c 1 i+j,c 2 j + k.c 3 k (23) 

In this form, with a vector to operate upon implied, the 
nature of c is fully exhibited. But the operand need not 


follow the operator. If it precedes it the result is just the 
same. Thus by (23), 

EC = Eucji + Ej.Cgj + Ek.c 3 k 

= c 1 E 1 .i + c 2 E 2 .j + c 3 E 3 .k = cE. . . (24) 

When, however, the operator is unsymmetrical, the vectors 
cE and EC are not identical. They are said to be conjugate to 
one another, so that in the symmetrical case of identity, c is 
also called a self-conjugate operator. The distinction between 
symmetrical and skew operators will appear later. 

There are six vectors concerned in c, viz., i, j, k, and ^i, c 2 j, 
c 3 k, which call I, J, K. Thus 

c = i.I+j.J + k,K ..... (25) 

is the type of a symmetrical operator referred to the principal 
axes. Now the general form of c referred to any axes and with 
the symmetrical restriction removed is obtained by turning 
these six vectors to any six others; thus, 

< = a.l + b.m + c.n, ...... (26) 

so that <E = a.lE + b.mE + c.nE J ..... (27) 

E( = Ea.l + Eb.m + Ec.n, ..... (28) 

show the linear function and its conjugate explicitly. This is 
(with a changed notation, however) Prof. Gibbs's way of re- 
garding linear operators. The arrangement of vectors in (26) 
he terms a dyadic, each term of two paired vectors (a . 1, &c.,) 
being, a dyad. Prof. Gibbs has considerably developed the 
theory of dyadics. 

Returning to the simpler form (25) or (22), we may note the 
effect of the performance of the operation symbolised by c a 
number of times, directly or inversely. Thus by (22) or (21), 
or (20), we have 


and so on. Thus c n means the operator whose principals are 
the nth powers of those of the primitive c, with the same 
axes. We therefore get a succession of ellipsoids with similarly 


directed principal axes, only altering their lengths, starting 
from initial E with constant tensor. (That is, provided the 
principal c's are all positive. If some be negative, we have 
other surfaces of the second order to consider.) 

We may also manipulate in the same simple manner any 
number of operators, a, b, c, &c., which have the same principal 
axes, though they may differ in other respects. Thus, using 
the same kind of notation, 

aE = i . a 1 E l + j . a 2 E 2 + k . a 3 E 3 , \ 

bdE = abE = i . a^Ej + j . a 2 6 2 E 2 + k . a 3 6 3 E 3 , I . (30) 

c&aE = a&cE = i . a^c^ + j . a 2 6 2 c 2 E 2 + k . a^c^, J 

and so on. That is, the successive action of any number of 
symmetrical operators with common principal axes is equiva- 
lent to the action of a single operator of the same kind, whose 
principals are the products of the similar principals of the set 
of operators. The resulting ellipsoids have their axes parallel 

Next, multiply the equation (20) by any other vector F, 

FcE = c 1 E 1 F 1 + c 2 E 2 F 2 + c 3 E 3 F 3 . . . . (31) 

By symmetry we see that this is identically the same as 
EcF. We might also conclude this from the fact that cE and 
EC are the same vector ; or, thirdly, by forming cF, and then 
multiplying it by E. We may therefore regard EcF as the 
scalar product of E and cF (or Fc), or, as the equal scalar pro- 
duct of EC (or cE) and F. This reciprocity is the general cha- 
racteristic of symmetrical operators, by which they can be 
distinguished from the skew operators. The electrical mean- 
ing in terms of displacement and electric force is that the com- 
ponent displacement in any direction N, due to an electric force 
acting in any other direction M, equals the component dis- 
placement along M, due to an electric force with the same 
tensor acting along N. 

Theory of Displacement in an Eolotropic Dielectric. The 

Solution for a Point-Source. 

163. We see from the preceding that when it is the electric 
force that is given in a dielectric there is no difficulty in find- 
ing the displacement ; and conversely, when the displacement 


is given, the electric force similarly becomes known through 
the values of the principal permittivities. Suppose, however, 
that the data do not include a knowledge of either the electric 
force or the displacement, both of which have to be found to 
suit other data. It is then sufficient to find either, the linear 
connection settling the other. Thus, to take an explicit case 
which has, for a reason which will -appear, a special interest, 
suppose the dielectric medium to have uniform permittivity 
transverse to the axis of k, but to be differently permittive to 
displacement parallel to this axis. Further, put a point-charge 
q at the origin, and enquire what is the equilibrium distribu- 
tion of displacement. 

The statement that there is a charge q at the origin means 
nothing more than that displacement diverges or emanates from 
that place, to the integral amount q. We also understand, of 
course, that the displacement has no divergence anywhere else. 
But the manner of emanation is, so far, left quite arbitrary. 
Certain general results may, however, be readily arrived at. To 
begin with, if the medium is isotropic, the emanation of the 
displacement must not favour one direction more than another, 
from which it follows that the displacement at distance r from 
the source is spread uniformly over the area 47rr 2 , so that q/lirr 2 
expresses its density. Now if we alter the permittivity to dis- 
placement parallel to k only, we favour displacement in that 
direction if the permittivity is increased, at the expense of the 
transverse displacement, because the total amount, which 
measures the strength of the source, is the same. Conversely, 
if we reduce the permittivity parallel to k, we favour the 
transverse displacement. The lines of displacement, originally 
spread equably, therefore separate themselves about the k axis 
(both ways), and concentrate themselves transversely, or about 
the equatorial plane, the plane passing through the charge 
which is cut by the k axis perpendicularly. 

If we carry this process so far as to reduce the permittivity 
along k to zero, there can be no displacement at all parallel 
to k, so that the displacement must be entirely confined to 
the equatorial plane itself. In this plane it spreads equably 
from the source, because of the transverse isotropy assumed at 
the beginning. The amount q therefore spreads uniformly over 
the circle of circumference 2?rr in the displacement sheet, so 


that the density (linear) is now q/2irr. The law of the inverse 
square of the distance has been replaced by the law of the 
inverse distance, with confinement to a single plane, however. 

Next, if we introducs planar eolotropy, say by reducing the 
permittivity in direction j, we cause the displacement in the 
sheet to concentrate itself about the i axis, at the expense of 
the displacement parallel to j. Finally, if we abolish 
altogether the permittivity parallel to j, the whole of the 
displacement will be confined to the i axis, half going one 
way and half the other. 

We may go further by introducing heterogeneity. Thus, 
reduce the permittivity on one side of the equatorial plane. 
The effect will be to favour displacement on the other side j 
and, in the limit, when the permittivity is altogether abolished 
on the first side, the whole of the displacement goes unilaterally 
along the k axis on the other side. 

In the case of the planar distribution of displacement, the 
surface-density is infinite, but the linear-density is finite, except 
at the source. We may, however, spread out the source along 
a finite straight line (part of the k axis) when the displacement 
will simultaneously spread out in parallel sheets, giving a finite 
surface-density. Similarly, in the case of the linear distribu- 
tion of displacement along the i axis, we may spread out the 
source upon a portion of the j, k plane, when the former line 
of displacement will become a tube, with finite surface-density 
of displacement inside it. 

In the above manner, therefore, we obtain a general know- 
ledge of results without mathematics. Except, however, in the 
extreme cases of spherical isotropy, of planar isotropy with zero 
permittivity perpendicular to the plane, and of purely linear 
displacement due to the vanishing of the transverse permit- 
tivity, and other extreme cases that may be named, we do not 
obtain an exact knowledge of the results, except in one respect, 
that the total displacement must always be the same. Return, 
therefore, to the case of initial isotropy upset by changed per- 
mittivity parallel to the k axis, or, if need be, with three differ- 
ent principal permittivities, so that the displacement is 

cF, . . . (32) 
if F be the electric force. We know that in the state of equi- 


librium, there must be no voltage in any circuit, or the curl of 
P must be zero, so that F must be the slope - VP of a scalar 
P, the potential. Furthermore, the divergence of D is the 
electrification- density p. Uniting, then, these results, we obtain 

div(-cVP) = />, ..... (33) 

or (c 1 V 1 2 + c 2 V 2 2 + c 3 V 3 2 )P=- /0 , . . . (34) 

which is the characteristic of P when referred to the principal 
axes. Now when there is a point-source at the origin, and the 
three c's are equal, we know that the potential is q/^irrc at 
distance r. From this, remembering that r 2 is the sum of the 
squares of the components of r, it is easy to see that the 
potential in the case of eolotropy is 

where f is some constant, because this expression (35) satisfies 
(34) with p = 0, that is to say, away from the source. 

The constant / may be evaluated by calculating the displace- 
ment passing through the surface of any sphere r = constant, 
according to (35), and equating it to q. The result is /= (c^Cg)"*. 
The complete potential in the eolotropic medium due to the 
point-source q at the origin is therefore 

g/47T ( } 


The equipotential surfaces are therefore ellipsoids centred at 
the charge, the equation of any one being 

. . . (37) 

so that the lengths of the principal axes are proportional to the 
square roots of the principal permittivities. The lines of elec- 
tric force cut through the equipotential surfaces at right angles. 
But the lines of displacement do not do so, on account of the 
eolotropy. To find their nature, first derive the electric force 
from the potential by differentiation. Then (36) gives 


and the displacement is obtained by operating by c on this, 
which cancels the c~\ and gives 



The displacement is therefore radial, or parallel to r itself. 
That is, the lines of displacement remain straight, only altering 
their distribution as the medium is made eolotropic. 

Observe, in passing, that the scalar product FD varies as 
P 4 ; that is, the density of the energy varies as the fourth 
power of the potential. Also note that Fr is proportional to P ; 
that is, the radial component of F, or the component parallel 
to the displacement, varies as P/r. 

If we select any pair of equipotential surfaces between which 
the electric force and displacement are given by (38), (39), we 
may, if we please, do away with the rest of the electric field. 
That is, we may let the displacement and electric force 
terminate abruptly upon the two ellipsoidal surfaces. What 
is left will still be in equilibrium. For the voltage remains 
zero in every circuit possible. This is obviously true between 
the surfaces, because no change has been made there. It is also 
true in any circuit beyond the surfaces, because of the absence 
of electric force. And lastly, it is true for any circuit partly 
within and partly beyond the region of electric force, because 
the electric force is perpendicular to the surface. So the electric 
field is self-contained. The electrification is on the surfaces, 
where the normal component of the displacement measures the 
surface-density. That is, there is a total electrification q on 
the inner and - q on the outer surface. The elastance of the 
condenser, or " leyden," to use Lord Rayleigh's word, formed 
by the two surfaces, is the ratio of the voltage between them 
to q. That is, the coefficient of q in (36) is the elastance be- 
tween any equipotential surface and the one at infinity. 

In thus abolishing the electric field beyond the limited 
region selected, we may at the same time change the nature of 
the medium, making it isotropic, for example,- or conducting, 
&c. Any change is permissible that does not introduce 
sources that will disturb the equilibrium of the electric field 
between the equipotential surfaces. 

Starting from isotropy, when the equipotential surface is a 
sphere, if we keep Cj = c 2 constant, and reduce c 3 , the sphere 
will become an oblate spheroid, like the earth (and other 
bodies) flattened at the poles. Let the common value of c t 


and c 2 be denoted by c , then the potential due to the central 
charge q is 

P s __ Qfa _ , (40) 

and the displacement and electric force are given by 

As c 3 is continuously reduced the oblate spheroid of equili- 
brium becomes flattened more and more, and is finally, when 
c 3 = 0, reduced to a circular disc. The displacement is now 
entirely in the equatorial plane, so that if we terminate the 
displacement at distance r from the centre, we obtain a 
circular line of electrification (that is, on the edge of the disc). 
We may have the displacement going from the central charge 
to an equal negative charge spread over the circle, or from one 
circle to another in the same plane ; and so on. 

Similarly, when c x and c 2 are unequal, but c 3 = 0, the circular 
disc is replaced by an elliptical disc. But, without introducing 
transverse eolotropy, we see that the abolition of the permittiv- 
ity in a certain direction has the effect of converting the usual 
tridimensional solutions relating to an isotropic medium to 
bidimensional solutions, in which the displacement due to 
sources situated in any plane perpendicular to the axis of 
symmetry is also in that plane. We may have any number of 
such planar distributions side by side, with the axis of symmetry 
running through them. But they are quite independent of 
one another. If we restore the permittivity parallel to the 
axis of symmetry, the displacement will usually spread out to 
suit tridimensional isotropy. The exception is when every 
plane perpendicular to the axis has identically similar sources, 
similarly situated. Then the restoration of the permittivity 
will produce no change. 

Theory of the relative Motion of Electrification and the 

Medium. The Solution for a Point- Source in steady 

rectilinear motion. The Equilibrium Surfaces in General. 

164. Let us now pass to what is, at first sight, an entirely 

different problem, but which involves essentially the same 

mathematics. As before, let there be a point-source of displace- 


ment at the origin, but let the dielectric medium be homo- 
geneous and fully isotropic. The displacement will be radially 
distributed, without bias one way or another. Now, keeping 
the charge fixed in space, imagine the medium to move steadily 
from right to left past the charge. How will this affect the 
displacement ? 

If we desire the solution expressing how the displacement 
changes from its initial distribution, as the medium is brought 
from rest into steady motion, we can obtain it by going the 
right way to work. But it is complex, and will, therefore, not 
be given here, especially as it is not intelligible without close 
study. But if we only wish to know the finally-assumed dis- 
tribution of displacement when the initial irregularities have 
subsided, we may obtain and express the result in a compara- 
tively simple manner. Thus, to begin from the foundation, we 
have the two fundamental circuital laws, 

curl(H-h) = cpE + /ou, .... (42) 
curl(e-E) = fipH, (43) 

where E and H are the electric and magnetic forces, /t and c 
the inductivity and permittivity, p the density of electrification, 
supposed to have velocity u ; and e, h are the motional electric 
and magnetic forces given by 

e = VwB, h = VDw, . . . (44) 

B and D being the induction and displacement, and w the 
velocity of the medium. (See equations (1), (2), or (3), (4), 
66, and equations (5), (6), 44, for the motional forces. Also 
33 to 70 generally.) 

Now in our present case, E and H are steady, and the elec- 
trification is at rest. The right sides of (42), (43) are 
therefore ze/o, which is a great simplification. By (42) and 
the second o~ (44) we obtain 

curl H = curl h = curl VDw. . . . (45) 
But B is circuital, and therefore so is H. It follows that 

H = VuD, (46) 

if we substitute - u for w, for future convenience. This gives 
H explicitly in terms of the displacement, when that is known, 
so we have no further trouble with the magnetic force. 


To find the displacement, (43) gives, along with the first of 

curl(E + VuB) = 0, .... (47) 
or say, curlf=0, if f=E + VuB. . , , (48) 

Now put the value of B in f by (46), thus, 

. . (49) 

since /zc 2 = l, v being the speed of propagation of distur- 
bances, and u the tensor of u, supposed to be parallel to k. 

Expanding by the fundamental formula (52), 114, we 

. . (50) 

Now let f = (1 - tt 2 /v 2 )F. Insert in (50), and divide by 
(1 - tt 2 /^ 2 ). The result is 


This vector F must also have no curl. Therefore, if P is the 
potential whose slope is F, we have 

d? dP n u? } dP 
%=-_-, J^ 2 = - .b = -(!-) 

ax ay v* dz 

and the components of displacement are given by 

*& %' *&' < 53 > 

if Ci = C2 = Cj an( j c 3 = c(l-u 2 /v 2 ). . . (54) 

Now observe that (53) express the displacement in terms of the 
electric force in an eolotropic medium at rest, whose principal 
permittivities are given by (54), when P represents the electro- 
static potential. Our present problem of a moving isotropic 
medium is therefore reduced to one relating to a stationary 
eolotropic medium. Thus, to state the comparison fairly, when 
the isotropic medium moves bodily past a stationary charge, 
the displacement distribution becomes identically the same as if 
the medium were at rest, but had its permittivity in the direc- 


tion of motion reduced from c to c(l - w 2 /v 2 ). The solution is 
therefore given by (40), (41) in 163, by taking c = c, and c 3 
as in (54). The displacement concentrates itself about the 
equatorial plane, or plane through the charge perpendicular to 
the axis of motion in the one problem, or axis of reduced per- 
mittivity in the other. In the limit, when the speed of motion 
reaches v, the speed of light, the displacement is wholly planar, 
as in the eolotropic case when the permittivity c 3 vanishes. 

Observe, in (52), that the square of u occurs. It does not 
matter, therefore, which way the motion takes place, so far as 
the displacement is concerned. But it makes a great difference 
in the magnetic force which accompanies the lateral concen- 
tration of the displacement. Being given by (46), we see that 
H reverses itself when the direction of motion is reversed. 

It should be carefully noted that P, which is the electrostatic 
potential in the eolotropic stationary medium, is not the electro- 
static potential in the moving medium, although it is the same 
function precisely. In the eolotropic medium its slope is the 
electric force F. In the moving medium its slope is the same 
force F, but this is no longer the electric force, which is E 
instead, obtained from F by (51) or (52). Again, whilst the 
displacement and electric force are not parallel in the stationary 
eolotropic medium ; on the other hand, in the moving medium 
they are parallel, because the medium is isotropic. To com- 
plete the differences, there is no magnetic force in the eolotropic 
case, but there is in the moving medium, and it is its existence 
which allows the parallel to exist as regards the identical 
distributions of displacement. 

Instead of moving the medium past a fixed charge, we may 
fix the medium, and move the charge through t the reverse 
way at the same speed. This is why we put w = - u. The 
above results apply when the charge q moves with velocity u 
through the stationary medium. But it is necessary now to 
travel with the charge in order to see the same results, because 
the origin is taken at the charge, and it is now in motion. The 
preliminary mathematics is therefore less easy. 

The above theory of convection- currents and the eolotropic 
comparison I have given before.* I have now to add a some- 
what important correction. For the opportunity of making 
* "Electrical Papers," Vol. II., pp. 492-499, 504-516. 


this correction I am indebted to Mr. G. F. C. Searle, of Cam- 
bridge, who has been at the trouble of working his way through 
my former calculations. Whilst fully confirming my results 
for a point- charge, and therefore all results obtained there- 
from by integration, he has recently (in a private communica- 
tion) cast doubt upon the validity of the extension of the 
solution to the case of a moving charged conducting sphere> 
and has asked the plain question (in effect) : What justification 
is there for taking distributions of displacement calculated in 
the above manner, and cutting them short by surfaces to which 
the displacement is perpendicular, which surfaces are then 
made equilibrium surfaces? For example, in the solution for a 
point- charge, I assumed that the solution was the same for a 
charged sphere, because the lines of electric force met it per- 
pendicularly. On examination, however, I find that there is no 
justification for this process. The electric force should not be 
normal to a surface of equilibrium, exceptions excepted. 

The true boundary condition for equilibrium, however, needs 
no fresh investigation, being implicitly contained in the above. 
In the stationary case of eolotropy, the vector F must have no 
curl, and since it is the electric force, it is the same as saying 
that there must be no voltage in any circuit, so that F must 
be normal to an equilibrium surface. Now, in the problem 
of a moving medium with charge fixed the same condition for- 
mally obtains, viz., that F shall have no curl. But as it is not 
the electric force E, it is clear that the displacement (which is 
parallel to E) cannot be perpendicular to an equilibrium sur- 
face. The voltage calculated by the fictitious electric force F 
must come to zero in every circuit. The boundary condition 
is, therefore, that F is perpendicular to an equilibrium surface. 
That is, P = constant is the equation to a surface of equilibrium 
(no longer equipotential in the electrostatic sense), where P is, 
however, the same function as in the eolotropic stationary 
problem of electrostatics. 

Therefore, by 163, we see that the sphere of equilibrium 
when the medium (isotropic) is stationary, becomes an oblate 
spheroid when the medium is set into a state of steady motion, 
the shorter axis being parallel to the line of motion. In the 
limit, when the speed is v, the spheroid is flattened to a 
circular disc, on whose edge only is the electrification. 



This applies when the whole dielectric medium is moving 
past a fixed spheroidal surface of electrification. It equally 
applies when the electrification moves the other way through a 
fixed medium. There will be no electric force in its interior. 
We may therefore fill it up with conducting matter, provided 
we do not interfere with the free motion of the external 
medium through it in the one case, or with the rest of the 
external medium when the conductor moves through it. But 
in the limiting case of motion at the speed of light, when only 
a single circular line of electrification is concerned, it would 
seem to be immaterial whether the conductor be a flat disc or 
a sphere, subject to the reservation of the last sentence. 

Whether it is possible to move matter through the ether 
without disturbing it forms an entirely different question, to 
which no definite answer can be given at present. We can, in 
any case, fall back upon the more abstract theory of electrifi- 
cation moving through the ether, without having conductors 
to interfere. 

Theory of the relative Motion of Magnetification and the 


165. From the preceding, relating to the magnetic effects 
produced by moving the medium bodily past stationary elec- 
trification ; or equivalently, by moving the electrification the 
other way through the medium ; or, more generally, by rela- 
tive motion of the sources of displacement and the medium 
supporting it, we may readily deduce the corresponding results 
when the sources are of the flux induction, instead of displace- 
ment. Thus, in (42), (43), do away with the convective electric 
current /ou, and substitute convective magnetic current cm, 
where o- is volume-density of magnetification (suppositional). 
Our circuital equations are then 

curl(H-h) = cpE, . .... (55) 
curl (e-E) = /*pH + <ni, . . . (56) 

with the auxiliary equations of motional electric and magnetic 

e = VwB, h = VDw, .... (57) 

as before, where w is the velocity of the medium, and with the 


further auxiliary conditions that the divergence of the displace- 
ment is zero, whilst that of the induction is or. Now we really 
do not need to employ the formal mathematics. Knowing the 
results relating to the motion of electrification, we may infer 
the corresponding ones concerning the motion of magnetifica- 
tion, by making use of the analogies between the electric and 
magnetic sides of electromagnetism, translating results in the 
appropriate manner. But as our object is to illustrate the 
working of vectors as well as electromagnetic principles, it 
will not be desirable to leave out the mathematics entirely, 
especially as the safe carrying out of the analogies requires 
that they should be thoroughly understood first, which requires 
some practice. 

First, if we convert the problem to one of stationary waves, 
by keeping the sources at rest, and letting only the medium 
move bodily past them, we have, when the stationary state of 
induction and displacement is reached, disappearance of the 
right members of (55) and (56). Then (56) gives, when united 
with the first of (57), 

curl E = curl e = curl VwB. . . . (58) 

In the result for a moving point-charge, curl E is perpendicular 
to w, so now we may write 

..... (59) 

where w = - u, because, similarly, curl H will be perpendicular 
to w, and so make VBu circuital. Here u is the equivalent 
velocity of the sources, when the medium is stationary. Thus 
E, and therefore D, are fully known in terms of B. Comparing 
with (46), we see that the electric force set up when a magnetic 
charge moves is related to the induction from the charge in 
the same way as the magnetic force set up when an electric 
charge moves is related to the displacement from it, with, 
however, a change of sign, or reversal of direction. 

Using the second of (57) in the other circuital equation (55) 

curl(H-VuD) = = curlf, say, . . (60) 

where, by (59), 

. . (61) 



or, if u is parallel to k, 

f = H + VkVkH = H + (kH 8 - H) 

. . (62) 

from which we see that the vector F derived from f by 
f=(l -u 2 /v 2 )T has no curl, or is derived from a scalar poten- 
tial fl thus, F = - vQ, and that the induction is given by 

*"' B *=-"S' B 3 =-,d-wf. (63; 

That is, the induction distributes itself in the same way as if 
the medium were at rest, but had its inductivity reduced from 
p to /*(! -u?/v z ) in the line of motion. In fact, the theory of 
the effects produced by relative motion of magnetic sources 
and the medium is essentially the same as that of moving 
electric sources ; translating displacement in the latter case to 
induction in the former, permittivity to inductivity, electric 
potential (real or fictitious, as the case may be) to magnetic 
potential, and the magnetic induction which accompanies 
moving electrification to electric displacement set up by 
moving magnetification, remembering, however, the change 
of sign. 

We know, therefore, the result of the steady rectilinear 
motion of any distribution of magnetic sources. If the same 
motion is common to all the elementary sources, we reduce the 
problem to that of magnetic eolotropy without relative motion 
of the medium and sources, and there is a definite system of sur- 
faces of equilibrium (ft = constant) where we may, if we please, 
terminate the electric and magnetic fields, thereby substituting 
new arrangements of magnetic sources for the old. In this 
case it is most convenient to keep the sources at rest, and 
move the medium alone. 

But should the relative motion of source and medium not be 
the same for every source, then, since we cannot move the 
medium bodily more than one way at a time, we may imagine 
it to be stationary, and obtain the effect due to the motion of 
the magnetification by superimposing the separate effects of 
the elementary sources, which are known by the above. 


Theory of the relative Motion of Magnetisation and the 
Medium. Increased Induction as well as Eolotropic 

166. There is, however, no such thing as magnetifi cation, 
the magnetic analogue of electrification. The induction is 
always circuital. If it were not circuital, we should have 
unipolar magnets. We must, therefore, somewhat modify the 
conditions assumed to prevail in the above, in order to come 
closer to reality. Consider, therefore, instead of magnetifica- 
tion, a distribution of intrinsic magnetisation. This quantity 
is I = /xh , where h is the equivalent intrinsic magnetic force. 
Abolish o- in (56), and introduce h in (55). The circuital 
equations are now 

curl (H - VDw - h ) = cpE, . . . (64) 

curl (VwB - E) = /^H, .... (65) 

where we have introduced the motional forces (57). In the 
steady state, the right members vanish as before. Now assume 

E = VwB, (66) 

so that the theory is unchanged so far, the displacement 
depending on the induction and velocity of the medium in the 
same way as when the source was magnetification. We shall 
see the limitation of application later. Next put (66) in (64) 
and we find, if the fraction 1 u 2 /v 2 be denoted by s, 

curl f= curl h , (67) 

curl F = curl h /s, .... (68) 

where f and F are the same vectors (in terms of H) as in tho 
last case, such that sF = f. We see that the vector F has now 
definite curl, and that the induction is derived from F by 

B^/iFp B 2 = ^F 2 , B 3 = MsF 3 ; . (69) 

that is to say B = AF, where A is the inductivity operator whose 
scalar principals are /x, p, and ps. We therefore have to find 
the steady state from these complete connections, 

divB = 0, B = AF, curl F = curl h /s . (70) 

That is, the induction is the same as in a stationary eolotropic 
medium (according to A), provided the intensity of the intrinsic 


magnetic force be increased from h to h /s. This is an 
important point. 

We know that there are certain respects in which the theory 
of induction due to intrinsic magnetic force is identical with 
that due to magnetification. For instance, the steady induction 
outside a magnet due to h is the same as that due to a dis- 
tribution of cr, measured by the convergence of the intrinsic 
magnetisation. From this we might hastily conclude that the 
disturbance from the steady distribution produced by moving 
the medium would be the same for h as for the equivalent o-, 
provided we keep outside the region of magnetisation. But 
the above investigation only partly confirms this conclusion. It 
shows a likeness and a difference. The likeness is in the 
eolotropic peculiarity brought in by the motion. In both cases 
we may do away with the motion provided we simultaneously 
reduce the inductivity parallel to the (abolished) motion from 
fj. to s/x, so as to cause the induction to retreat from this direc- 
tion and concentrate itself transversely. The difference is in 
the reckoning of the strength of sources. In the case of 
magnetification (as of electrification) we do not alter the 
strength of the source. But in the case of magnetisation we 
do, or, at any rate, produce an equivalent result. For, along 
with the reduction of inductivity in the direction of motion, 
we require to increase the intensity of the intrinsic magnetic 
force from h to h /s. Remember that s is a proper fraction, 
going from unity to zero as the speed of motion increases fron. 
to v t the speed of propagation of disturbances. 

To exemplify this, put 

..... (71) 

as we see we may do, by the third of (70). Using this in the 
second and first of (70) we obtain 

.... (72) 
the equation of the potential 12. Or, in terms of Cartesians, 

XV + v 2 2 + v 8 8 )fl -/*-Woi + v 2 A 02 + *VA S ) . (73) 

The equivalent magnetification is the convergence of Ah /s, 
not of fihQ the real intensity of magnetisation, nor yet of /*h /s, 
the same increased in a constant ratio. 


There is, therefore, a remarkable difference between the two 
cases of motion of the medium parallel to and transverse to 
the lines of real magnetisation. Thus, first let h be parallel 
to w or k. Then the s" 1 outside the brackets in (73) cancels 
the s inside, so that the right member becomes simply /xV 3 ^ 03 , 
where A 03 is now the tensor of h . That is, the effective 
magnetification is the convergence of the real magnetisation, 
just as when the medium is at rest. But if h is perpendicular 
to k, say parallel to i, the right member of (73) becomes 
/xs-'V^ou or t* 16 effective magnetification is s~ l times the con 
vergence of the magnetisation, and is therefore increased. 

Suppose, for example, our magnet is a straight filamentary 
magnet. It has two poles, of strength m and - m say, so that 
induction to the amount m diverges from one and converges to 
the other pole, whilst continuity is made between them in the 
filament itself. Now let the medium move past the filament, 
the direction of motion being parallel to it. Then the strength 
of the poles is unchanged, but the induction outside is diverted 
laterally by the effective reduced inductivity parallel to the 
filament. This case, then, resembles that of a pair of oppositely 
signed point-charges, if we keep outside the filament. In the 
limit, therefore, when the speed is raised to v, we have a pair 
of parallel plane induction-sheets whose cores are joined by the 
straight filament of induction, to make continuity. 

For (66) to be valid, B should be such as to make 
circuital. This requires that w and curl B should be perpen- 
dicular to one another. They are perpendicular in the example 
just mentioned, on account of the symmetry of B with respect 
to the axis or line of motion. But if the same filament be held 
transversely across the line of motion the property stated is no 
longer true, so that (66) is not true, and this will, to an 
unknown extent, upset the later equations. We can, however, 
still employ them provisionally to obtain a first approximation 
to the results. On this understanding, then, the strength of 
the poles (effective) is made m/s and - m/s, with unimpaired 
effective inductivity parallel to the filament, whilst that per- 
pendicular to the filament is reduced, as before. So there is 
now an increased total induction as well as concentration 
about the plane through the filament perpendicular to the 


We may, however, arrange matters in such a way that the 
equation (66) shall be still valid when the line of motion is 
perpendicular to the magnetisation, and so obtain an exact 
solution showing the increased induction, that is, exact in the 
absence of working errors. Let the region of magnetisation be 
confined between two infinite parallel planes, and the magneti- 
sation I = fih be uniformly distributed, parallel to the 
boundaries. We thus do away with the poles. Now when the 
medium is at rest the induction is B = I within the plate 
(which may be of any thickness), and zero outside, whilst 
there is no displacement. The intrinsic magnetic force h pro- 
duces the greatest effect that it can produce unaided. But if 
the medium be made to move steadily straight across the 
magnetised region, then, after certain transient effects have 
passed away (which may be readily calculated, because they 
form simply plane electromagnetic waves), although there will 
still be no induction (or displacement) outside the plate, that 
within it will be increased (without change of direction) from 
I to I/s. That is, the motional magnetic force comes in to 
assist the intrinsic magnetic force. Here we have the explana- 
tion of the previous result relating to the increased effective 
strength of poles. 

Along with this increased induction, there will be electric 
force, according to (66). It is entirely within the magnetised 
region. As the speed increases up to v t the induction and the 
accompanying displacement in the plate go up to infinity. 

The theory of an electrised plate is similar. The displace- 
ment due to intrinsic electrisation J = ce , where e is the 
equivalent electric force, the electrisation being uniform and 
parallel to the sides of the plate, will be increased from D = J 
to D = J/s by setting the medium moving straight through it, 
it being now the motional electric force that assists the in- 
trinsic. The accompanying magnetic force is accordingly given 

Outside the plate there is no disturbance in the steady state. 

It will, of course, be understood that if, instead of supposi- 
tional impressed or intrinsic forces in a uniform medium, we 
employ actual material plates, magnetised or electrised as the 
case may be, they must, in the first place, be non-conductors, 


and next, they must not interfere with the supposed uniform 
motion of the medium. In short, the reservation is similar to 
that mentioned in connection with moving charged conductors. 
Of course, in our present case, the plates may be moved, 
whilst the medium is supposed to be at rest. 

It will be readily seen that the full investigation of the 
effects of moving practicable electromagnetic arrangements of 
conductors presents considerable difficulties. We can, how- 
ever, get some information relating to the motion of a linear 

Theory of the Relative Motion of Electric Currents and 
the Medium. 

167. Thus, let us first abolish the intrinsic magnetisation 
of 166, and substitute equivalent electric current. Here, 
by equivalent electric current, we mean a distribution of 
electric current which produces the same induction as the 
intrinsic magnetisation ; so that if both were to exist together, 
the induction would be everywhere doubled ; and then, if either 
of them were negatived, the resulting induction would be nil. 
That this is possible, whatever may be the distribution of 
inductivity, is a very remarkable property. The induction due 
to magnetisation is conditioned solely by the curl of the intrin- 
sic magnetic force, and the " poles " are essentially quite a 
secondary matter. That is, we may vary the poles as we like, 
provided we do not alter the curl of the intrinsic force, without 
affecting the induction or the associated energy. 

The question then presents itself whether this equivalence 
continues to hold good when the medium is set in motion past 
the stationary magnetisation or electric current respectively. In 
the circuital equation (64), put h = 0, and introduce a term 
on the right side, producing 

curl(H-VDw) = C + cpE, . . . (75) 

expressing the first circuital law ; whilst (65), or 

curl(VwB-E) = /ijpE .... (76) 

expresses the second. Now (75) only differs from (64) in the 
substitution of C here, for curlh there. That is, the new 
C and the old curlh are equivalent. We may, therefore, 


dismiss the idea of magnetisation, and let the source of the 
induction be any distribution of intrinsic electric current C . 
With it is associated a certain distribution of induction, iden- 
tical with that due to any distribution of intrinsic magnetisation 
for which we have curlh = C , and the equivalence persists 
when the medium is moving, or when the sources are moving. 
Corresponding to (70) we shall have 

divB = 0, B = AF, curlF = C /s, . (77) 
for the determination of the induction in the eolotropic manner 
previously pursued, when the steady state of affairs is reached j 
subject also to the previous reservation that w and curl B are 
perpendicular. That is to say, the induction is affected by the 
motion in the same way as if the medium had its inductivity 
decreased from /x to fts in the direction of motion, without any 
change transversely, whilst at the same time the source C is 
effectively increased to C /s. 

Equations (77) are suitable when the electric currents are in 
planes perpendicular to the direction of motion. For instance, 
let there be (to take a very easy example) two parallel plane 
sheets of electric current of surface-density C and - C respec- 
tively. The induction between them, which is B=//,C when 
at rest, becomes ftC /s when they are in motion with velocity u 
perpendicular to their planes. Here s is the fraction (1 - u 2 /v*) 
as before, and the electric force accompanying -the changed in 
duction is simply the motional electric force. 

There is another way of making the eolotropic comparison, 
namely, by employing the vector f instead of F. Then, instead 
of (77) we shall have 

divB = 0, B = (A/)f, curlf=C . . (78) 
In this way of looking at the matter we regard as suffering 
no change of effective strength, whilst the eolotropic operator 
X/s is such as to indicate increased inductivity (from p to /A/*) 
transverse to the motion, and unchanged inductivity parallel 
to it. The vector f is the excess of the magnetic force H of 
the flux over the motional magnetic force. Inasmuch as this 
way emphasises the fact that the intrinsic sources are really 
constant, it is not without its advantages. But the idea of 
increased transverse inductivity is rather an unmanageable one 
in general, especially in the case of moving electrification. 


Jf the sources and the medium have a common uniform 
translational motion, it may be seen from the general circuital 
equations that the steady distribution of the fluxes with respect 
to their sources is unaffected by the motion. That is, if we 
travel with the medium there is no change observable. This 
applies in the case of circuital sources (as curlh above), as 
well as divergent sources (as of electrification). The natural- 
ness of the result is obvious, when the relativity of motion is 

The General Linear -Operator. 

168. It is now necessary to leave these special cases of 
eolotropy for fear of being carried away too far from the main 
subject, which is, the nature of linear vector operators, in ter- 
mination of this chapter on vector analysis. Up to the present 
only the symmetrical operator has been under consideration. 
Three rectangular axes are concerned, each of which is identi- 
fied with parallelism of the force and flux, or of a vector and a 
linear function thereof. Now, if we associate the algebraical 
ratio of the force to the flux with the three directions of 
parallelism, we see that the symmetrical linear operator depends 
upon three vectors. If they were arbitrary, this would involve 
nine scalar specifications ; but, being coperpendicular, there 
are really only six independent specifications ; and, moreover, 
if we transform to any other system of axes (independent or 
non-coplanar) there can still be no more than six independent 

But it is easy to see that the general linear vector operator 
must involve nine scalar specifications, viz., three for each of 
the three independent axes of reference that may be chosen. 
Thus, let D be any linear function of E. We must then have, 
in terms of the i, j, k components, the following set of equa- 
tions : 

..... (79) 

= C 31 E 1 + C 32 E 2 + C 33 E 3>. 

where the nine c's may have any values we please. The axes 
of reference are perpendicular only for convenience ; any set of 
three independent axes may be used, with nine properly deter- 


mined c's to match. The relation between the vectors D and 
E, which is fully exhibited in (79), may be conveniently 
symbolised by 

D = cE, ...... (80) 

where c is the general linear operator. 

If we exchange c 12 and c 21 , &c., in the set (79) we shall 
usually change D, of course. Let it become D'. It is then 
fully given by the set 


and these relations between D' and E may be symbolised, like 
(80), by the single equation 

D' = c'E ....... (82) 

The manipulation of sets of equations like (79) and (81) is 
lengthy and laborious. By the use of the linear operators, 
however, with proper attention to the laws governing them, 
the vectors may be manipulated with facility. Thus, to give 
the first and easiest example that presents itself, we may 
instantly turn AB to Ace^B. This is obvious enough, inas- 
much as the operation ii dicated by c" 1 will be precisely can- 
celled by the operation c, so that cc -1 B = B. But there is much 
more in it than that. For ActT^B is not merely the scalar pro- 
duct of the vectors A and cc~ 1 B, that is, of A and B, but is 
the scalar product of the vectors Ac and c~ 1 B, which are quite 
different. The vector Ac is the same as the vector c'A, whilst 
c -1 B is the vector which, when operated upon by c, gives the 
vector B. 

The constituents of the inverse operator c" 1 may be found 
by solution of (79). Since D is a linear function of E, it follows 
that E is a linear function of D ; or the E's may be expressed 
in terms of the D's by means of a set of equations like (79), 
with new coefficients instead of the c's there. Similarly as 
regards the inverse operator c'" 1 belonging to (81) and (82). 

The vector formed by taking half the sum of the vectors D 
and D' is a symmetrical function of E, say 

AK . . ... (83) 


The constituents of X with two equal suffixes, namely, c n , 
C 22 C 33 are tne same as those of c and of c. But the other 
constituents are half the sum of the corresponding ones of c 
and c' ; for instance J(c 12 + c 21 ) in place of c 12 and c 21 . From 
this we see that the operator A is its own conjugate, or is self- 
con jugate, making AE and A/E identical. 

On the other hand, the vector formed by taking half the 
difference of D and D' is a simple vector product. By inspec- 
tion of (79) and (81), and remembering the structure of a 
vector product, we may see that 

where a is the vector given by 

. (85) 

We see, therefore, that any linear function, if it be not 
already of the symmetrical kind, may be represented as the 
sum of a symmetrical function and of a vector product. Thus, 
by addition and subtraction of (83) and (84) we obtain 

cE = D = AE + VaE, ..... (86) 

c'E = D' = AE-VaE ..... (87) 

That is, in terms of the operators alone, 

c = A + Va, ..... (88) 

c' = A-Va, ..... (89) 

where c (and therefore c') is general, whilst A is symmetrical. 

The vector a is quite intrinsic, and independent of axes of refer- 
ence. The symmetrical operator A involves six scalars, the vector 
a three more, thus making up the full nine. 

Notice, by (86), (87), that 

.... (90) 

or the scalar product of the force and the flux does not involve 
the vector a at all. 

The Dyadical Structure of Linear Operators. 

169. Now go back to the equations (79). Observe that the 
right members, being the sums of three pairs of simple products, 


are themselves scalar products. Thus, let v&ree new vectors 
defined by 

be introduced. Then we see that equations (79) are simply 
D^^E, D 2 = c 2 E, D 3 = c 3 E, . . (92) 

and, therefore, by combining them, we produce the one vector 

D = i.c 1 E+j.c 2 E + k.c 3 E, . . . (93) 

thus proving that every linear operator may be exhibited in 
the dyadical form 

c = i.c 1 +j.c 2 + k.c 3 , .... (94) 

whether it be symmetrical or not. The conjugate operator is, 

c' = i.c' 1 +j.c' 2 + k.c' 3 , .... (95) 

where the accented vectors may be obtained from the un- 
accented given in (91) by changing c 12 to c n , <fec., just as D' 
was got from D, in fact. 

It is now easily to be proved that 

cE = Ec', Ec = c'B, , . . (96) 

that is, if we wish to remove an operator from before to behind 
a vector, we may do so by simply turning the operator to its 
conjugate ; that is, by putting on the accent, or by removing 
it if already there. It will, of course, be understood that when 
a, vector E follows c, and we use the dyadical form of c, as in 
(94), it is only the three second vectors that unite with E \ 
whereas, if E precedes c, it is the first vectors in c that unite 
with it. Thus : 

Ec = Ei.c 1 + Ej.c 2 + Ek.c 3 , . -. . (97) 

c'E = i.c' 1 E+j.c' 9 E + k.c' 3 E. . . . (98) 

These are identical vectors always. In the symmetrical case, 
the accents may be dropped, of course ; then EA, = A.E. It will 
be seen that the above way of writing and working 
dyadics fits in precisely with all the previous notation em- 
ployed in the vector algebra. The dots in (94) are merely 


separators, not signs of multiplication. When we put on a 
vector, before or behind, it unites with the vectors it is not 
separated from by dots ; we therefore obtain the forms (97), 
(98), in agreement with the principles of notation described in 
die early part of this chapter. 

We have already pointed out that EAT, where A, is sym- 
metrical, may be regarded as the scalar product of E and AT, 
or of EA, (the same as AE), and T. The corresponding general 
property is the same ; that is, c being general, EcT is the 
scalar product of EC and F, or of E and cP. But EC is the 
same as c'E, and cP the same as Fc', so we have 

EcF = c'EF = EFc' = Fc'E = FEc. . . (99) 

The operator c must be united with one or other of the two 
vectors E and F, but it is immaterial which it is, provided we 
always change to the conjugate properly, as exemplified. 

The same applies when there are many linear operators. 
Let there be three, a, 6, c. Then we have, to illustrate how 
the transformations are made, 

. (100) 

and so on. Here the brackets are introduced to show the 
association of the vectors and the operators. 

We may also introduce anywhere a new operator, accom- 
panied, of course, by the reciprocal operator ; thus 

cE[aF] = cB[db- l bf] = eE[a&- 1 (F&')] = &c. . (101) 

The above will be sufficient to show the great power gained 
by the use of the operators, enabling one to do almost at sight 
algebraical work which would, in Cartesians, cover pages. 

Hamilton's Theorem. 

170. The following little theorem, due to Hamilton, will 
serve to illustrate the working of vectors, as well as another 
purpose which will appear later. 

Let m and n be a pair of vectors, then their vector product 
Vmn is perpendicular to both, by our definition of a vector pro- 
duct. That is, 

= mVmn, = nVmn, . . . (102) 


by our definition of a scalar product, and the parallelepipedal 
property. Now introduce cc~ l betwen the V's and the vectors 
preceding them in (102). Thus 

= mcc- 1 Vmn, = ncc- 1 Vmn. . . (103) 

These assert (and we cannot but believe it) that the vector 
c^Vmn is perpendicular to the vectors me and nc, or c'm and 
c'n ; it is, therefore, parallel to their vector product. The last 
statement is expressed by 

c- 1 Vmn = Vc / mc'n, .... (104) 

where x is an unknown (or so far undetermined) scalar. Or, 
operating on both sides by c, 

a;Vmn = cVc'mc'n. .... (105) 

To find the value of x, we have merely to multiply (105) by a 
third vector, say 1. For this gives 

zlVmn = IcVc'mc'n = c'lVc'mc'n, (106) 

which gives the value of x explicitly, namely, 

< 107 > 

to be used in (105), which, with x thus settled, is the state- 
ment of the little (but important) theorem. The quantity x 
appears to depend upon 1, m, n. These vectors, however, enter 
into both the numerator and denominator in such a way that 
they can be wholly eliminated from x. This may be seen by 
expanding the numerator and denominator. In fact, a; is a 
pure constant, depending upon the operator c alone. 

In the symmetrical case we may find its value by taking 
1, m, n to be i, j, k, and the latter to be the principal axes. 

cl = ci = Cji, c'm = cjj, c'n = c 3 k, 

if c lf c 2 , c 3 are the principal scalar c's. So, by (107), 


the continued product of the principal c's. 
In the general case we may put, as we know, 


Using this in (107), on the understanding that c lt c 2 , c 3 are 
the principals of A now, we shall get, similarly, 

.... (109) 

which it will be a useful exercise to verify. 

Since c is any linear operator, (105) remains true when for c 
we substitute its conjugate c', producing 

aj'Vmn = c'Vcnicn, . . . . (110) 

where x is got from the expression for x by changing c' to its 
conjugate c. But c and c only differ in the changed sign of a, 
as in (88), (89). On the other hand (109) contains a quadrati- 
cally, so that a reversal of its sign makes no difference. There- 
fore, x' is the same as x, which might not have been anticipated 
at the beginning of the evaluation. 

Hamilton's Cubic and the Invariants concerned. 

171. The reader of Prof. Tait's profound treatise on 
Quaternions will probably stick at three places in particular, 
to say nothing of the numerous minor sticking-points that 
present themselves in all mathematical works of any value, 
and which may be readily overcome by the reader if he be 
a real student as well. First, there is the fundamental 
Chapter II., wherein the rules for the multiplication of vectors 
are made to depend upon the difficult mathematics of spherical 
conies, combined with versors, quaternions and metaphysics. 
Next, Chapter IV., where the reader may be puzzled to find out 
why the usual simple notion of differentials is departed from, 
although the departure is said to be obligatory. Thirdly, 
Chapter V., where the reader will be stopped nearly at the 
beginning by a rather formidable investigation of Hamilton's 
cubic. Only when the student is well acquainted with the 
nature of linear operators and how to work them can he tackle 
such an investigation. He should, therefore, pass on, to obtain 
the necessary experience. On then returning to the cubic he 
may find the investigation not so difficult after all, especially if 
it be simplified by some changes calculated to bring out 
the main points of the work more plainly. 

The reader is led to think that the object of the investiga- 
tion is to invert a linear operator that is, given D = cE, to find 



E = c~ 1 D. But if this were all, it would be a remarkable 
example of how not to do it. For the inversion of a linear 
operator can be easily effected by other far simpler and more 
natural means. The mere inversion is nothing. It is the 
cubic equation itself that is the real goal. The process of 
reaching it is simplified by the omission of inverse operations. 
(It is also simplified by not introducing the auxiliary function 
called x-) 

The fundamental cubic is derived from equation (105), or 
the equivalent equation (110) last investigated. I should 
remark that this equation is frequently useful in advanced 
vector-analysis as a transformation formula, turning Vcmcn to 
a function of Vmn. Now use the form (110), or 

#Vmn = c'Vcmcn, ...... (HI) 

where a; is a known function of c, viz., 


Here c is any linear operator. Now if g be a constant, gm is 
a linear function of m, and therefore (c - g)m is any linear 
function of m; that is, c-g is a general linear operator. 
Equation (111) therefore remains true when we substitute 
c - g for c in it, not forgetting to make the change in x, and also 
the equivalent change in c', viz., from c' to c' g. Equation 
(111) then becomes 

x g Vmn = (c' - g)V(e - g)m(c - g)n, (113) 

where x g is what x becomes by the change, that is, by (112), 

x - (c-<7)lV( c 


Remembering that g is a constant, we may readily expand (113) 
to the form 

= c'Vcmcn - g (Vcmcn + c Vmcn + c'Vcmn) 
- # 3 Vmn +g' 2 (c'Vmn + Vmcn + Vcmn), . . (115) 

where the cubic function of g on the left side is the expansion 
of x g . The new coefficients x l and rc >2 are, of course, to be 


found by expanding (114), but we do not want their values 
immediately. Since a; is a function of c only, x ff is a function 
of c and g only, so that x l and X 2 are functions of c only. 

Now since (111) is an identity, so is (115) ; and since g may 
have any value, the last equation must be identically true for 
every power of g concerned, taken one at a time. That is, it 
splits into four identities. ~Now, on comparing the coefficients 
of g* on the left and right sides, we obtain (111) again. Simi- 
larly, we observe that the coefficients of g 5 are the same. So 
far, then, we have nothing new. But the g and g 2 terms give 

. . (116) 
. . (117) 

which are fresh identities. From these various others may be 
deduced by elimination or combination. The one we are seek- 
ing is obtained by operating on the first by c' and on the second 
by c' 2 , and then subtracting the second result from the first. 
This eliminates the vector in the brackets and leaves 

- a? 2 c' 2 )Vmn = c'Vcmcn - c' 3 Vmn, . (118) 
= (a;-c' 3 )Vmn, . . . (119) 

where the transition from (118) to (119) is made by usin& 
(111) again. Rearranging, we have the final result, 

which is Hamilton's cubic. Note that Vmn may be any vector 
we please, and may, therefore, be denoted by a single letter. 
Observe, also, that the cubic function of c' is of the same form 
as that of g in the expansion of x. 

Since (120) is true for all linear operators, we may write c 
instead of c', giving 

x-x lC + x 2 c*-c* = ..... (121) 

This form is what we should have arrived at had we started 
from (105) instead of (110). The value of x is the same in 
either case, and it may be inferred from this that x l and x 2 do 
not suffer any change when we pass from a linear operator to 
its conjugate. 

That the complex function x g is independent of 1, m, n, may 
be seen by inspecting (114), and observing the parallel epipedal 



form of numerator and denominator. For, suppose we alter 
1 to 1 + am, where a is any constant. The addition made to the 
numerator vanishes by the parallelepipedal property, and simi- 
larly in the denominator. The same invariance obtains when 
we change 1 to 1 + an. Also, the change of 1 to any scalar 
multiple of itself alters both the numerator and denominator 
in the same ratio. But, unless 1, m, n are coplanar, we may 
turn 1 to any vector by adding to it vectors parallel to 1, m 
and n of the right size. Therefore, x g is the same whatever 
vector 1 may be. By the same reasoning m may be any vector, 
and so may n. But 1, m, n should be independent vectors (that 
is, not coplanar). This is not because we can suppose that an 
actual discontinuity occurs when (for example) 1 is brought 
into the plane m, n, but merely because the expression x 9 
assumes an indeterminate form in the coplanar case. 

From the invariance of x ff follows that of the three functions 
it contains, namely, x, x v and x y That of #, of course, may be 
independently seen by itself, readily enough, but that of the others 
is less plain, because on expansion they are found to each 
involve three parallelepipedal products in the numerator instead 
of only one. Thus, expanding (114) we obtain 

1 Vcmcn 4- m Vend + n Vclcm / 1 o o \ 

- -' ' ' (122) 

cl Vmn -f cmVnl + en Vim / 1 o Q \ 

* 2== lVmn~ 

In these we may, by the above, substitute c' for c, that is, 
turn the linear operator to its conjugate. This does not mean 
that x g is independent of the vector a which comes in when 
the operator is not symmetrical, but that it only involves a 

The value of x being 

x = CjC^ + aca, . . . . . (124) 

expressed in terms of the principal constants of the symmetrical 
operator and the rotation vector, we may obtain the reduced 
values of x l and X 2 from it directly instead of from their general 
jxpressions. Thus : turn c to c-g (in 124), making it 

$)* ( 125 ) 


On expansion, the coefficients of g and # 2 show that 

X l = C 2 C 3 + C 3 C 1 + C 1 C 2 + ft2 > (126) 
Z 2 = Cj+C 2 + C 3 . 

Thus x z is independent of a, whilst #j involves its square, as 
we concluded previously. 

When referred to the principal axes, Hamilton's cubic there- 
fore reduces to 

= foc^ + aca) - (c^ + c 2 c 3 + CgCj + a 2 )c 

*<';*><} (127) 

and in the symmetrical case of vanishing a to 

Here the axes are coperpendicular. But we may disregard 
the principal axes altogether, and write the general cubic in 
the form 

where the g'a are the roots of x g = 0. If, then, these roots are 
all real and different, there are three directions of parallelism 
of r and cr, or three r's such that 

i=0A <**=9f9 <x z =y z i y (130) 

Now multiply the first by r 2 and the second by T V giving 

W-ftVi. ricr 2 =5' 2 r 1 r 2 . . . (131) 

The left members are not usually equal, but in the sym 
metrical case they are, and then the right members are 
equalised. Then 1^ = 0, or r x is perpendicular to r 2 . Simi- 
larly r 2 is perpendicular to r 3 . This is the case of three 
mutually perpendicular axes of parallelism of force and flux 
we started from. 

The Inversion of Linear Operators. 

172. If we write c~ l for c in the cubic (121) we see that 
c -1 r becomes expressed as a function of r, cr, and c 2 r. This is 
one way of inverting the operator, but a very clumsy way. 
The simple way is in terms of dyads, and is fully described by 
saying that if 

c = a.l + b.m + c.n . ,. . . (132) 


is any linear operator in dyadical form, then its reciprocal is 
c- 1 = L.A + M.B + N.C, . . . (133) 

where A, B, C is the set complementary to a, b, c, and L, M, N 
the set complementary to 1, m, n. We have already used 
these complementary vectors in the early part of this chapter, 
equations (54) and (55), 114. We showed by elementary 
considerations that any vector r could be expressed in terms of 
any three independent vectors a, b, c by 

r = rA.a + rB.b + rC.c, . . . (134) 
where the complementary vectors A, B, C are got from a, b, c 


A = I^, B = ^i, 0-U (135) 

aVbc aVbc aVbc v ' 

These equations serve to define the set complementary to a, b, c. 
We also showed at the same time that the vector r could be 
expressed in terms of the complementary set by 

r = ra.A + rb.B + rc.O. .;-..' . (136) 

To verify, we have merely to multiply (134) by A, B, C in turn, 
and (136) by a, b, c in turn. 

These equations may be written 

r = (a.A + b.B + c.C)r, . . . (137) 
r = (A.a + B.b + C.c)r, . ,, . (138) 

showing that the dyadic in the brackets is of a very peculiar 
kind, inasmuch as its resultant effect on any vector is to repro- 
duce the vector. That is, taken as a whole, and disregarding 
its detailed functions, the dyadic is equivalent to unity. 
Now, suppose it is given that 

R = (a.l + b.m + c.n)r, . . . (139) 
so that R is any linear function of r. We know that 

R = (a.A + b.B + c.C)R, . . . (140) 

and that r=(L.l + M.m + N.n)r, . . . (141) 

by the property of the complementary vectors explained 
Comparing (140) with (139), we see that 

lr = AR, mr = BR, nr = CR. . . (142) 


Using these in. (141), we convert it to 

r = (L.A + M.B + N.C)R, . . . (143) 

comparing which with (139), we see that the inversion of the 
dyadic has been effected in a simple and neat manner. 

Professor Gibbs calls the vectors A, B, the reciprocals of 
a, b, c. They have some of the properties of reciprocals. Thus, 

aA = l, bB = l, cO = l. . . . (144) 

But it seems to me that the use of the word reciprocal in this 
manner is open to objection. It is in conflict with the obvious 
meaning of the recipocal a" 1 of a vector a, that it is the vector 
whose tensor is the reciprocal of that of a with unchanged ort 
(or with ort reversed in the quaternionic system). It will also 
be observed that Gibbs's reciprocal of a vector depends not upon 
that vector alone, but upon two others as well. It would 
seem desirable, therefore, to choose some other name than re- 
ciprocal. I have provisionally used the word " complementary " 
in the above, to avoid confusion with the more natural use of 

Skew Product of a Vector and a Dyadic. The Differentia- 
tion of Linear Operators. 

173. In connection with dyadtcs, it should be remarked 
that we have only employed them in the manner they usually 
present themselves in physical mathematics, namely, so as to 
make direct products with the vectors they are associated with. 
But there is also the skew product to be considered in a com- 
plete treatment. Thus, 

. . (145) 
. (U6) 

Observe that, as before with the direct products, the vector 
r only combines with the vectors nearest to it in the dyadic, 
and not separated from it by the dots. Notice, too, that 
whereas the direct product <r of the dyadic </> and a vector 
r is a vector ; on the other hand, the skew products V<r 
and Vr< are themselves dyadics, and behave as such in 
union with vectors. Thus sV<r is a vector, and so is sVr<, 
the first being got by making scalar products of s with a, b, c, 


and the second with Vra, &o. But to go further in this 
direction would be to go beyond the scope of the present 

The differentiation of linear operators must, however, be 
mentioned, because the process is of frequent occurrence in 
electromagnetic investigations. If we think only of the 
differential coefficient of a scalar, that it is its rate of increase 
with some variable, it might seem at first sight that the 
differential coefficient of a linear operator was nonsense. But 
a little consideration will show that it is a perfectly natural, 
and by no means a difficult conception. Thus, from 

D = cE, ...... (147) 

where c is a linear operator, we obtain, by differentiation with 
respect to a scalar variable, say the time, 


Here in the second term on the right we suppose that c is con- 
stant, and in the first that E is constant. The meaning 
of cE, then, is the rate of increase of D when c alone varies. It 
is the linear operator whose constituents are the rates of 
increase of the constituents of c. That this is so will be evident 
on differentiating the set of equations (79). Therefore, if 

c=a.l + b.m + c.n, .... (149) 
we shall have 

. . (150) 

the sum of two dyadics. But it may be that our axes of 
reference are invariable, as for example when 

c = i.c 1 +j.c 2 + k.c 3 , .... (151) 

i, j, k being a fixed set of rectangular orts. Then we have the 
one dyadic 

.. (152) 

where, of course, the dots over the i and j do not signify any 

In an isotropic dielectric of variable permittivity the electric 
stress leads to the force - V c (^cE 2 ) per unit volume, where the 
scalar c alone is differentiated, so that the result is - JE 2 Vc. 


But if the dielectric be eolotropic, the corresponding force is 
This is equivalent to 

< 153 > 

where dc/dx, &c., are differential coefficients of c as above ex- 
plained. It means the same as 

-(V D -V E )(iED), .... (154) 

where V D means that D alone, and V E that E alone is 

Summary of Method of Vector Analysis. 

174. In the last paragraph I came dangerously near to 
overstepping the imposed limits of my treatment of vectors, 
which is meant to present the subject merely in the form it 
issumes in ordinary physical mathematics. If we were to 
ignore the physical applications, and treat vector algebra as a 
branch of pure mathematics, regardless of practical limitations, 
there would be no bounds to the investigation of the subject. 
It will now be convenient to wind up with a few remarks on the 
previous, and on the nature and prospects of vector analysis. 

Since we live in a world of vectors, an algebra or language 
of vectors is a positive necessity. At the commencement of 
this chapter, 97 to 102, 1 made some general remarks on the 
nature of cartesian analysis, vector analysis, and quaternions ; 
and the reader is recommended to read them again from a more 
advanced point of view. Then, he was supposed to know next 
to nothing about vectors. Now, although he need not have 
absorbed all the special applications of vector algebra that have 
been given since, it may be presumed that he has acquired a 
general knowledge of the principles of the subject. There is 
no longer any question as to the desirability and utility of 
vectorial analysis. The present question is rather as to the 
form the vector algebra should take. On this point there is 
likely to be considerable difference of opinion, according to the 
point of view assumed, whether with regard to physical appli- 
cations, or abstract mathematical theory. Let us then, to begin 
with, summarise the leading points in the algebra given above. 

First, there is the idea of the vector as a distinct entity. 
Nearly everyone nowadays knows and apprecktes the idea. 


And it is a noteworthy fact that ignorant men have long been 
in advance of the learned about vectors. Ignorant people, like 
Faraday, naturally think in vectors. They may know nothing 
jf their formal manipulation, but if they think about vectors, 
ihey think of them as vectors, that is, directed magnitudes. 
No ignorant man could or would think about the three com- 
ponents of a vector separately, and disconnected from one 
another. That is a device of learned mathematicians, to enable 
them to evade vectors. The device is often useful, especially 
for calculating purposes, but for general purposes of reasoning 
the manipulation of the scalar components instead of the 
vector itself is entirely wrong. 

In order to facilitate the reading of vector work, the vector 
has its special type, thus E. This Clarendon, or any very similar 
neat black type (not block letters), is meant to mark the vector, 
always the vector, and never anything else, with the obvious 
exception of headlines that speak for themselves. Also, to 
economise letters, and ease the strain on the memory, the same 
letter suitably modified serves for the tensor, the ort, and the 
three scalar components. Thus E is the tensor, or size, and Ej 
the ort (signifying the orientation) so that E = EE X ; whilst the 
scalar components referred to rectangular axes are Ej, E 2 , E 3 . 
The physical dimensions may be most conveniently merged in 
the tensor E. Every vector has its species. The one we are 
most familiar with is the space vector, or straight line joining 
two points, or more strictly the displacement from one point to 
another. But all kinds of vector magnitudes are formally 
similar in having size and ort, and therefore, when considered 
vectorially, obey the same laws. The properties of the space 
vector therefore supply us with the rules or laws of vector 

The addition and subtraction of vectors is embodied in the 
assertion (whose truth is obvious) that the sum of any number 
of vectors making (when put end to end) a circuit is zero ; or, 
in another form, all the various paths by which we may pass 
from one point to another are vectorially equivalent, namely, to 
the vector straight from point to point. Every vector equation 
therefore expresses this fact. Every term is a vector ; and if 
all the vectors be put on one side, with zero on the other, we 
express the circuital property ; whilst if we have vectors on both 



sides of the equation, we assert the vectorial equivalence of two 

Any vector A and its reciprocal A" 1 have the same ort, and 
their tensors are reciprocal to one another. 

Coming next to combinations of vectors of the nature of 
products, we find they are of two kinds, scalar and vector. The 
scalar product of and F is denoted by EF, and the vector 
product by VEF. Their tensors are EFcos0 and EFsinfl 
respectively, 6 being the included angle. The scalar product 
is directionless ; the ort of the vector product is that of the 
normal to the plane of E and F. The tensor and ort of VEF 
are V EF and VjEF. 

The justification for the treatment of the scalar and vector 
products as fundamental ideas in vector algebra is to be found 
in the distributive property they possess, thus, 

., . (155) 

which hold good for any number of vectors. In the first 
equation the manipulation is as in common algebra, in all re- 
spects. In the second, in all respects save one ; for the re- 
versal of the order of the vectors in a vector product negatives 
it, thus, Vac = - Vca. 

Here we have very complex relations of geometry brought 
down to elementary algebra. As Prof. Gibbs has remarked, 
the scalar and vector product, which consist of the cosine and 
sine of trigonometry combined with certain other simple notions, 
are "incomparably more amenable" to algebraical treatment 
than the sine and cosine themselves. 

When we go on to combinations in threes, fours, &c., we find 
it is the same thing over again in various forms. Thus c.ab 
and cVab and VcVab define themselves by the above, being ab 
times c, and the scalar and vector products of c and Vab re- 

Similarly with four vectors, as in dc.ab, the product of dc 
and ab ; dc.Vab, which is dc times Vab ; d.cVab, which is cVab 
times d ; dVcVab and VdVcVab, which are the scalar and vec- 
tor products of d and VcVab ; VdcVab arid VVdcVab, which 
are the scalar and vector products of Vdc and Vab. Com- 
binations of two vectors are universal, and those of three are 


pretty frequent, but those of four are exceptional. But it 
should be noted that the scalar and vector product of two 
vectors are involved throughout, no new idea being introduced 
when more than two vectors are concerned. 

When we pass on to analysis, we find just the same vector 
algebra to be involved. Vectors are differentiated with respect 
to scalars to make new vectors in the same way as scalars, and 
the ideas connected with differentials (infinitesimal) and diffe- 
rential coefficients are essentially the same for vectors as for 
scalars. We never require to differentiate a vector with respect 
to a vector, and there is a very good reason for it, because the 
operation is an indeterminate one. 

The fictitious vector y is omnipresent in tridimensional ana- 
lysis. It only differs from a vector in being a differentiator as 
well, so that it follows vector rules combined with other func- 
tions. With it are associated the ideas of the slope of a scalar, 
and the divergence and curl of a vector, with corresponding 
important theorems relating to the transition from line to sur- 
face, and from surface to volume summations. The theory of 
potentials in its broad sense is also involved in the mathematics 
of y, including the relations, direct and inverse, of potential, 
slope, curl, and divergence. 

As for the linear operator, that is only the scalar and vector 
product system again in a special form. Professor Gibbs's 
dyadic is a useful idea, and I have modified his notation to suit 
the rest. 

It is neither necessary nor desirable that a student should 
know all that is sketched out above before he makes practical 
use of it. For vector analysis, like many other things, is best 
studied in the concrete application. The principles may be 
very concisely stated, being little more than explanations of 
the scalar and vector product, with some conventions about 
notation. A student might learn this by heart, and be little 
the better for it. He must sit down and work if he wants to 
assimilate it usefully. He may work with scalar products 
only to begin with, for quite a large ground is covered thereby. 
When familiarised with the working of vectors by practice so 
far as the scalar product goes, the further introduction of the 
vector product will come comparatively easy. The former 
knowledge is fully utilised, and also receives a vast extension 


of application. For the vector product is a powerful engine, 
which, with its companion the scalar product, is fully capable 
of working elaborate mathematical analysis in a concise and 
systematic manner. The student need not trouble about linear 
operators at first. He will grow into them. It is far more 
important that he should understand the operation of y as a 
vector, and its meaning in the space variation of functions. 

Unsuitability of Quaternions for Physical Needs. Axiom : - 
Once a Vector, always a Vector. 

175. Now, a few words regarding Quaternions. It is 
known that Sir W. Rowan Hamilton discovered or invented a 
remarkable system of mathematics, and that since his death 
the quaternionic mantle has adorned the shoulders of Prof. 
Tait, who has repeatedly advocated the claims of Quaternions. 
Prof. Tait in particular emphasises its great power, simplicity, 
and perfect naturalness, on the one hand; and on the other 
tells the physicist that it is exactly what he wants for his phy 
sical purposes. It is also known that physicists, with great 
obstinacy, have been careful (generally speaking) to have 
nothing to do with Quaternions ; and, what is equally remark- 
able, writers who take up the subject of Vectors are (generally 
speaking) possessed of the idea that Quaternions is not 
exactly what they want, and so they go tinkering at it, trying 
to make it a little more intelligible, very much to the disgust 
of Prof. Tait, who would preserve the quaternionic stream pure 
and undefiled. Now, is Prof. Tait right, or are the defilers 
right? Opinions may differ. My own is that the answer all 
depends upon the point of view. 

If we put aside practical application to Physics, and look 
upon Quaternions entirely from the quaternionic point of 
view, then Prof. Tait is right, thoroughly right, and Quater- 
nions furnishes a uniquely simple and natural way of treating 
quaternions. Observe the emphasis. 

For consider what a quaternion is. It is the operator which 
turns one vector to another. For instance, we have 

B=A.A- I B + B-A- I .AB 


identically. Or, 

B = (A- 1 B + V.VA~ 1 B)A. . . , (157) 

Here the operator in the brackets, say q, depends upon A and 
B in such a way that it turns A to B ; thus, B = qA.. It is a 
quaternion. It has a scalar and a vector part. The general 
type of a quaternion is 

? = w, + Va, (158) 

where w is any scalar, and a is any vector. Since we know 
the laws of scalar and vector products we may readily 
deduce the laws of quaternions should we desire to do so. 
But we shall find that the above notation, though so well 
suited for vectors, is entirely unfitted for displaying the 
merits of quaternions. To do this we must follow Hamilton, 
and make the quaternion itself the master, and arrange 
the notation and conventions to suit it, regardless of the con- 
venience of the vector and the scalar. The result is the 
singularly powerful algebra of quaternions. To give a notion 
of its power and essential simplicity we may remark that the 
product of any number of quaternions, p, q, r t s, say, in the 
order named, is independent of the manner of association in 
that order. That is, 

pqrs =p(qrs) = (pq) (rs) = (pqr)s. (1 59) 

This remarkable property is the foundation of the simplicity 
of the quaternionic algebra. It is uniquely simple. No algebra 
of vectors can ever match it, and Prof. Tait is quite right in his 
laudations from the quaternionic point of view. 

But when Prof. Tait vaunts the perfect fitness and natural- 
ness of quaternions for use by the physicist in his inquiries, 
I think that he is quite wrong. For there are some very 
serious drawbacks connected with quaternions, when applied 
to vectors. The quaternion is regarded as a. complex of 
scalar and vector, and as the principles are made to suit the 
quaternion, the vector itself becomes a degraded quaternion, 
and behaves as a quaternion. That is, in a given equation, 
one vector may be a vector, and another be a quaternion. Or 
the same vector in one and the same equation, may be a 
vector in one place, and a quaternion (versor, or turner) in 
another, This amalgamation of the vectorial and quaternionic 


functions is very puzzling. You never know how things will 
turn out. 

Again, the vector having to submit to the quaternion, leads 
to the extraordinary result that the square of every vector is a 
negative scalar. This is merely because it is true for quad- 
rantal versors, and the vector has to follow suit. The reciprocal 
of a vector, too, goes the wrong way, merely to accommodate 
versors and quaternions. 

And yet this topsyturvy system is earnestly and seriously 
recommended to physicists as being precisely what they want. 
Not a bit of it. They don't want it. They have said so by 
their silence. Common sense of the fitness of things revolts 
against the quaternionic doctrines about vectors. Nothing 
could be more unnatural. 

Are they even convenient 1 Not to the physicist. He is very 
much concerned with vectors, but not at all, or at any rate 
scarcely at all, with quaternions. The vector algebra should 
satisfy his requirements, not those of the quaternion. Let the 
quaternion stand aside. The physicist wants, above all, to 
have clear ideas, and to him the double use of vectors, as 
vectors and versors, combined with unnatural properties of 
vectors to suit quaternions, is odious. Once a vector, always 
a vector, should be a cardinal axiom. 

If the usual investigations of physical mathematics involved 
quaternions, then the physicist would no doubt have to use 
them. But they do not. If you translate physical investi- 
gations into vectorial language, you do not get quaternions ; 
you get vector algebra instead. Even Prof. Tait's treatise 
teaches the same lesson, for in his physical applications the 
quaternion is hardly ever concerned. It is vector algebra, 
although expressed in the quaternionic notation. 

Vectors should be treated vectorially. When this is done, 
the subject is much simplified, and we are permitted to arrange 
our notation to suit physical requirements. This is a very 
important matter. Most calculations are, and always will be, 
scalar calculations, that is, according to common algebra. The 
special extensions to tridimensional space involving vectors 
should therefore be done so as to harmonise with the scalar 
calculations, and so that only one way of thinking is required, 
instead of two discrepant ways, and so that mutual conversion 


of scalar aiid vector algebra is facilitated. The system which 
I expound I believe to represent what the physicist wants, at 
least to begin with. It is what I have been expounding, 
though rather by example than precept, since 1882. There is 
only one point where I feel inclined to accept a change, viz., to 
put a prefix before the scalar product, say Sab instead of ab, to 
balance Vab. Not that the S prefix is of any use in the algebra 
as I do it, but that mathematical writers on vectors seem to 
want to put the scalar and vector parts together to make a 
general or complete product, say Qab = Sab + Vab. The function 
Qab, however, does not occur in physical applications, which are 
concerned with the scalar and vector products separately. If I 
could omit the V as well as the S I would do so ; but some sign 
is necessary, and the V of Hamilton is not very objectionable. 

Gibbs denotes the scalar and vector products by a./3 and a x ft 
using Greek letters. In my experience this is not a good way 
when worked out. It is hard to read, and is in serious conflict 
with the common use of dots and crosses in algebra, which use 
has to be given up. I use dots in their common meaning, 
either as multipliers or separators. 

Prof. Macfarlane, who is the latest vectorial propagandist, 
denotes the scalar and vector products by cos ab and Sin ab. 
The trigonometrical origin is obvious. But, on this point, I 
would refer the reader to the equations (155) (156) above, and 
to the very trenchant remark of Prof. Gibbs which I have 
already quoted, as to the incomparably greater amenity of the 
scalar and vector product to algebraical treatment than the 
trigonometrical functions themselves. The inference is that 
vector algebra is far more simple and fundamental than trigo- 
nometry, and that it is a mistake to base vectorial notation 
upon that of a special application thereof of a more complicated 
nature. I should rather prefer sea ab and vec ab. Better still, 
Sab and Vab, with the understanding that the S may be 
dropped unless specially wanted. 

Macfarlane has also a peculiar way of treating quaternions, 
about which I will express no opinion at present, being doubtful 
whether, if the use of quaternions is wanted, the quater- 
nionic system should not be used, with, however, a distinction 
well preserved between the vector and the quaternion, by 
special type or otherwise. 


I am in hopes that the Chapter which I now finish may serve 
as a stopgap till regular vectorial treatises come to be written 
suitable for physicists, based upon the vectorial treatment of 
vectors. The quaternionists want to throw away the " cartesian 
trammels," as they call them. This may do for quaternions, 
but with vectors would be a grave mistake. My system, so far 
from being inimical to the cartesian system of mathematics, is 
its very essence. 



Action at a Distance versus Intermediate Agency. Contrast 
of New with. Old Views about Electricity. 

176. It has often been observed that the universe is in an 
unstable condition. Nothing is still. Nor can we keep motion, 
once produced, to a particular quantity of matter. It is diffused 
or otherwise transferred to other matter, either immediately or 
eventually. The same fact is observed in the moral and intel- 
lectual worlds as in the material, but this only concerns us so 
far as to say that it underlies the communication of knowledge 
to others when the spirit moves, even though the task be of a 
thankless nature. 

The laws by which motions, or phenomena which ultimately 
depend upon motion, are transferred, naturally form an import- 
ant subject of study by physicists. There are two extreme 
main views concerning the process. There is the theory of 
instantaneous action at a distance between different bodies 
without an intervening medium ; and on the other hand there 
is the theory of propagation in time through and by means of 
an intervening medium. In the latter case the distant bodies 
do not really act upon one another, but only .seem to do so. 
They really act on the medium directly, and between the two 
are actions between contiguous parts of the medium itself. But 
in the former case the idea of a medium does not enter at all. 
We may, however, somewhat modify the view so that action, 
though seemingly instantaneous and direct, does take place 
through an intervening medium, the speed of transmission 
being so great as to be beyond recognition. Thus, the two 


ideas of direct action, and through a yielding medium, may be 
somewhat harmonised, by being made extreme cases of one 
theory. For example, if we know that there is a yielding 
medium and a finite speed, but that in a certain case under 
examination the influence of the yielding is insensible, 
then we may practically assume the speed to be infinite. But 
although this comes to the same thing as action at a distance, 
we need not go further and do away with the medium alto- 

There is another way of regarding the matter. We may 
explain propagation in time through a medium by actions at a 
distance. But this is useless as an explanation, being at best 
merely the expression of a mathematical equivalence. 

Now consider the transmission of sound. This consists, 
physically, of vibratory motions of matter, and is transmitted 
by waves in the air and in the bodies immersed in it. The 
speed through air is quite small, so small that it could not even 
escape the notice of the ancients, who were, on the whole, 
decidedly unscientific in their mental attitude towards natural 
phenomena. Suppose, however, that the speed of transmission 
of sound through air was a large multiple of what it is, so that 
the idea of a finite speed, or of speed at all, did not present 
itself. We should then have the main facts before us, that 
material bodies could vibrate according to certain laws, and 
that they could, moreover, set distant bodies vibrating. This 
would be the induction of vibrations, and it might be explained, 
in the absence of better knowledge, by means of action at a 
distance of matter upon matter, ultimately resolvable into some 
form of the inverse square law, not because there is anything 
essentially acoustical about it, but because of the properties of 
space. Furthermore, we should naturally be always associating 
sound with the bodies bounded by the air, but never with the 
air itself, which would not come into the theory. A deep- 
minded philosopher, who should explain matters in terms of 
an intervening medium, might not find his views be readily 
accepted, even though he pointed out independent evidence for 
the existence of his medium physiological and mechanical 
and that there was a harmony of essential properties entailed. 
Nothing short of actual proof of the finite speed of sound 
would convince the prejudiced. 



Now, although there is undoubtedly great difference In detail 
between the transmission of sound and of electrical disturb- 
ances, yet there is sufficient resemblance broadly to make the 
above suppositional case analogous to what has actually hap- 
pened in the science of eleetromagnetism. There were con- 
ductors and non-conductors, or insulators, and since the finite 
speed of propagation in the non-conducting space outside con- 
ductors was unknown, attention was almost entirely concen- 
trated upon the conductors and a suppositional fluid which 
was supposed to reside upon or in them, and to move 
about upon or through them. And the influence on dis- 
tant conductors was attributed to instantaneous action 
at a distance, ignoring an intermediate agency. Again, 
a very deep-minded philosopher elaborated a theory to ex- 
plain these actions by the intermediate agency of a yielding 
medium transmitting at finite speed. But although in doing 
so he utilised the same medium for whose existence there was 
already independent evidence, viz., the luminiferous ether, 
and pointed out the consistency of the essential properties re- 
quired in the two cases, and that his electromagnetic theory 
made even a far better theory of light than the old, yet his views 
did not spread very rapidly. The old views persisted in spite 
of the intrinsic probability of the new, and in spite of the large 
amount of evidence in support of the view that some medium 
outside conductors, and it may be also inside them as well, but 
not particularly conducting matter itself, was essentially con- 
cerned in the electrical phenomena. The value and validity 
of evidence varies according to the state of mind of the judge. 
To some who had seriously studied Maxwell's theory the 
evidence in its favour was overwhelming; others did not believe 
in it a bit. I am, nevertheless, inclined to think that it would 
have prevailed before very long, even had no direct evidence of 
the finite velocity been forthcoming. It was simply a question 
of time. But the experimental proof of the finite speed of 
transmission was forthcoming, and the very slow influence of 
theoretical reasoning on conservative minds was enforced by 
the common-sense appeal to facts. It is now as much a fact 
that electromagnetic waves are propagated outside conductors 
as that sound waves are propagated outside vibrating bodies. 
It is as legitimate a scientific inference that there is a medium 


to do it in one case as in the other, and there is inde- 
pendent evidence in favour of both media, air for sound 
and ether for electrical disturbances. It is also so exces- 
sively probable now that light vibrations are themselves 
nothing more than very rapid electromagnetic vibrations, that 
I think this view will fully prevail, even if the gap that exists 
between Hertzian and light vibrations is not filled up ex- 
perimentally, through want of proper appliances provided 
some quite new discovery of an unanticipated character is 
not made that will disprove the possibility of the assumed 

Now, the immediate question here is how to propagate a 
knowledge of the theory of electromagnetic waves. If we 
could assume the reader to have had a good mathematical train- 
ing, that would greatly ease matters. On the other hand, it is 
hardly any use trying to do it for those who have no mathe- 
matical knowledge least of all for those anti-mathematical 
attackers of the theory of the wave propagation of electrical 
disturbances who show plainly that they do not even know in- 
telligibly what a wave means, and what is implied by its existence. 
But, between the two, I think it is possible to do a good deal in 
the way of propagation by means of a detailed exposition of the 
theory of plane waves, especially in dielectrics. By the use of 
plane waves, the mathematical complexity of waves in general 
in a great measure disappears, and common algebra may be 
largely substituted for the analysis which occurs in more ad- 
vanced cases. Most of the essential properties and ideas may 
be assimilated by a thinking reader who is not advanced in his 
mathematics, and what is beyond him he can skip. Besides 
this, the treatment of plane waves is itself the best preliminary 
to more general cases. 

The problems that present themselves by the artificial limi- 
tation to plane waves are often of an abstract nature. There 
are, however, some important exceptions; the most notable 
being that of propagation along straight wires, of which the 
theory is essentially that of plane waves, modified by the 
resistance of the wires. Before, however, proceeding to the 
details of plane waves, which will form the subject of this 
chapter (although it will be needless to altogether exclude con- 
nected matters), it will be desirable to prepare the mind by an 


outline of some general notions concerning electromagnetic 
waves, irrespective of their precise type. 

General Notions about Electromagnetic Waves. Generation 

of Spherical Waves and Steady States. 


177. Consider a n^n-conducting dielectric, to begin with. 
The two properties it possesses of supporting electric displace- 
ment and magnetic induction, which we symbolise by ju- and c, 
the inductivity and permittivity, are, independently of our actual 
ignorance of their ultimate nature, so related that the speed of 
propagation v depends upon them in the way expressed by the 
equation /xcv 2 = 1, or v = (/AC)-*. Thus, an increase either of the 
permittivity, or of the inductivity, lowers the speed of trans- 
mission. In transparent bodies we cannot materially alter the 
inductivity, but we can very considerably increase the per- 
mittivity, and so lower the speed. If on the other hand, we 
wish to have infinite speed, for some practical purpose of calcu- 
lation, we may get it by assuming either p = or else c = 0. 
In the latter case, for example, we destroy the power of sup- 
porting electric displacement, whilst preserving the magnetic 
induction. This is what is done in magnetic problems (the 
theory of coils, self and mutual induction) when we ignore the 
existence of electric displacement. On the other hand, in the 
theory of condensers connected up by inductionless resistances, 
and in the electrostatic theory of a submarine cable, it is the 
magnetic induction that is ignored, in a manner equivalent to 
supposing that /* = 0. In either case we have infinite speed or 
apparent instantaneous action. 

Now, consider some of the consequences of the property of 
propagation with finite speed. Let there be a source of dis- 
turbance at a point for simplicity. It may be either an electric 
source or a magnetic source. By an electric source we mean 
a cause which will, if it continue steadily acting, result in 
setting up a state of electric displacement, whilst a magnetic 
source steadily acting would result in a state of magnetic 
induction. In the former case there will be no magnetic 
induction along with the displacement, and in the latter case 
no displacement along with the induction. Now, if the speed 
of propagation were infinite in the former case of an electric 
source, by the non existence of p, the steady state would be 


set up instantly, and consequently all variations of the intensity 
of the source would be immediately and simultaneously accom- 
panied by the appropriate corresponding distribution of dis- 
placement in the dielectric. Similarly, with a magnetic source, 
if the speed be infinite by the non-existence of c, all variations in 
the source will be simultaneously accompanied by the distribu- 
tion of magnetic induction appropriate to the instantaneous 
strength of the source. 

Now do away with the artificial assumption made, and let 
both p and c be finite, and v therefore also finite. At the time 
t after starting a source, the extreme distance reached by the 
disturbance it produces is vt. That is to say, beyond the sphere 
of radius vt, whose centre is at the source, there is no disturb- 
ance, whilst within it there is. The wave-front is thus a spheri- 
cal surface of radius vt, increasing uniformly with the time. 
Along with this continuous expansion of the range of action 
there are some other things to be considered. The mere spread- 
ing causes attenuation, or weakening of intensity as the disturb- 
ance travels away from the source. Besides this, the intensity 
of disturbance is not the same at a given distance in all direc- 
tions from the source, and since the displacement and induction 
are vectors, their directions are not everywhere the same. They 
are distributed in the circuital manner, so that we have expand- 
ing rings or sheets of displacement or induction. Whether the 
source be of the electric or the magnetic kind, it produces both 
fluxes initially and when varying, and generally speaking to an 
equal degree as regards energy. This is a main characteristic 
of pure electromagnetic waves, a coexistence of the electric and 
magnetic fluxes with equal energies ; and if the source be in a 
state of sufficiently rapid alternating variation this state of 
things continues. But if the source, after varying in any way, 
finally become steady, the generation of electromagnetic dis- 
turbances will speedily cease, and the production of a steady 
state will begin round the source of displacement if the source 
be electric, and of induction if the source be magnetic whilst 
the previously generated electromagnetic disturbances will pass 
away to a great distance. In the end, therefore, we have 
simply the steady state of the flux corresponding to the 
impressed force, whilst the electromagnetic disturbances are 
out of reach, still spreading out, however t and in doing so, 


leaving behind them the outside portion of the residual steady 
flux due to the source, which steady flux, however, requires an 
infinite time to become quite fully established. 

We may take a specially simple case in illustration of the 
above general characteristics. Let the source of energy be 
contained within a small spherical portion of the dielectric, of 
radius a, and let it be of the simplest type, viz., a uniform 
distribution of impressed electric force within the sphere. We 
know by static considerations alone what the nature of the 
final displacement due to such a source is. It is a circuital 
distribution, out from the sphere on one half, and in on the 
other, connection being made through the sphere itself by a 
uniform distribution ; being, in fact, of the same nature as the 
induction due to a spherical body uniformly magnetised in a 
medium of the same inductivity as its own. Now, we can 
describe the setting up of the final steady state due to the 
source, when it is suddenly started and kept constant later, 
thus : At the first moment an electromagnetic wave is generated 
on the surface of the sphere, which immediately spreads both 
ways and becomes a spherical shell, whose outer boundary 
goes outward at speed v, whilst the inner boundary goes inward. 
At the time t = a/v, therefore, the disturbance fills the sphere 
of radius 2a, the centre being just reached. The steady state 
then begins to form, commencing at the centre and expand- 
ing outwards thereafter at speed . At the time t = 2a/v, 
therefore, we have the steady state fully formed within 
the sphere of radius a, whilst just outside it is an electro- 
magnetic shell of depth 2a. Up to this moment the im- 
pressed force has been continuously working not, indeed, 
in all parts of the sphere it occupies, but in all parts 
passed by the front of the inward wave from the beginning 
until the centre was reached, and after that, in the parts not 
occupied by the already formed steady state. Consequently, at 
the moment t = 2a/v, when the steady state is fully formed 
throughout the region occupied by the impressed force, the 
latter ceases to work. It has wholly done its work. The 
amount done is twice the energy of the final complete steady 
state, say 2U. The rest of the work is done by the electro- 
magnetic wave itself. For the subsequent course of events is 
that the fully-formed electromagnetic shell of depth 2a runs 


out to infinity, of course expanding on the way, and in doing 
so it leaves the steady state behind it. That is to say, it 
drops a part of its contents as it moves on, so that the steady 
state is always fully formed right up to the rear of the expanding 
shell during the whole of its passage to infinity. This shell is 
not a pure electromagnetic wave, with equal electric and 
magnetic energies. The magnetic energy is constant, of 
amount ^U, on the whole journey, but the electric energy 
is in excess. The excess is employed in forming the steady 
state, so that when the shell has reached a great distance it 
becomes appreciably a pure electromagnetic wave, having the 
amount U of energy, half magnetic, half electric, with a slight 
excess in the latter, to be later left behind in forming the 
remainder of the steady state. The energy U of the shell is 
wholly wasted if the dielectric be unbounded, for there is 
nothing to stop the transference to an infinite distance. 

If we wish to form the steady state without this waste of 
energy, we must bring the impressed force into action very 
gradually infinitely slowly, in fact. In this way the energy 
wasted in the very weak electromagnetic waves generated will 
tend to become infinitely small, and the work done by the 
impressed force will tend to the value U, the energy of the 
steady state. 

When the sphere of impressed force is not finite, but is 
infinitely small, so that it may be regarded as a special kind of 
point-source, we simplify matters considerably in some respects. 
For, by the above, the result is that the moment the source 
starts, say at full strength, the steady state also immediately 
begins to grow, so that at the time t it occupies the sphere of 
radius vt. Outside it there is no disturbance, but on its surface is 
an infinitely thin electromagnetic shell, which performs the same 
functions as the previous finite shell. That is, it lays down 
the steady state as it expands, and then carries out to infinity 
in itself as much energy as it leaves behind. 

Notice, however, this peculiarity, that if we had started with 
what appears at first sight to be the simpler problem, viz., a 
point-source, we should be quite unable to see and understand 
the functions of the electromagnetic shell on the extreme verge 
of the steady field. The reader may compare this case with 
that of the establishment of the steady state of displacement 


due to another kind of point-source, viz., electrification suddenly 
brought to rest at a point after previous motion at the speed of 
light, as discussed in 55. There is a perfect similarity as re- 
spects the uniform growth of the steady state. But the nature 
of the bounding electromagnetic shell is not the same. In the 
case of electrification it belongs to the zero degree of spherical 
harmonics ; in our present problem it is of the first degree. 

If, after keeping on the impressed force at a point for, say, 
an interval T, we suddenly remove it, this is equivalent to 
keeping it on, but with the addition of an impressed force which 
is the negative of the former. From this we see that the re- 
sult of putting on the impressed force for an interval T only is to 
generate a shell of depth VT, which runs out to infinity. 
Within this shell is the steady electric displacement due to the 
source appropriate to the instantaneous position of the shell. 
On its outer surface is an electromagnetic wave moving out to 
infinity, and generating or laying down the steady electric dis- 
placement as it goes, whilst on the inner surface is another 
wave running after the first, and undoing its effects. That is, 
it takes up the displacement laid down by the first wave, so 
that inside the inner wave (just as outside the outer) there is 
no disturbance. 

Intermittent Source producing Steady States and Electro- 
magnetic Sheets. A Train of S.H. Waves. 

178. Using the same kind of point-source, let it act inter- 
mittently and alternatingly ; say, on positively for an interval 
a, then off for an interval /2, then on negatively for an interval 
a, followed by off for an interval ft, and so on recurrently, like 
positive and negative applications of a battery with dead in- 
tervals. The result is, by the last case, to divide the spherical 
space occupied by the disturbances at a given moment into 
concentric shells, of depths va and v/3 respectively. In the 
latter is no disturbance, since they correspond to the dead in- 
tervals. In the former are the steady displacements appro- 
priate to their distance from the source, consecutive shells 
being positive or negative according to the state of the source 
when they started from it. On their boundaries are electro- 
magnetic sheets generating these steady states continuously on 
one side, and destroying them on the other. 


Next, do away with the dead intervals, so that the applied 
force is like that of a common reversing key, first positive 
of full strength, and then negative of full strength, but without 
any interval. The result is to completely fill up the sphere of 
disturbance, the successive shells of depth va containing steady 
fields alternately positive and negative being brought into 
contact, and only separated from one another by the electro- 
magnetic sheets. This may be considered rather an unexpected 
result. Our impressed force is periodic, but expresses the 
extremest form of variation, namely discontinuities, and is only 
expressible simple-harmonically by an infinite series of simple- 
harmonic forces of all frequencies from zero to infinity. But 
only when the impressed force varies is electromagnetic dis- 
turbance generated, so that when in the above manner we 
confine its variations to be momentary, the electromagnetic 
disturbances are also momentary, with the consequent result 
that we have our dielectric occupied by quite steady states of 
displacement separated by infinitely-thin electromagnetic shells. 
The latter travel. The steady states also appear to travel, but 
do not really do so. They are being continuously generated and 
destroyed at their boundaries, but are quite steady elsewhere. 

If the impressed force vary simple-harmonically instead of 
discontinuously, the result is less simple. We find that there 
is no clean separation into steady states and electromagnetic 
waves, the two being mixed inextricably. We can see this by 
substituting for the simply periodic variation of the force a 
very great number of constant forces of small duration and of 
different strengths, so as to.roughly imitate the simple- harmonic 
variation. Every discontinuity in the force will behave in the 
way described, and the final result of the superimposition of 
effects, when we proceed to the limit and have continuous 
simple-harmonic variation of the force, is a train of simple- 
harmonic waves proceeding from the source. They are not 
pure electromagnetic waves, and there is no steady state any- 
where. The wave-length X is given by A = VT, where T is the 
period of the impressed force, or the reciprocal of the 
frequency. The electric and magnetic forces are simple-har- 
monic functions of the time everywhere, right up to the 
extreme wave-front of the disturbance, where, of course, is 
an electromagnetic sheet following a different law. But 


should the source occupy a finite spherical space, then 
the bounding sheet becomes a shell, of depth equal to the 
diameter of the sphere, as before described. Now the steady 
fields become insensible at a great distance in the discontinuous 
case ; to correspond with this we have the fact that the simple 
harmonic waves tend to become pure as they expand. The 
greatest departure from the pure state is round about the 
source. Here the frequency becomes a matter of importance. 
When very low, the wave-length is great, and we should, there- 
fore, have to go a great distance to find any sign of wave- 
propagation. Moreover, the waves themselves would be 
exceedingly weak. Only round the source is there sensible 
disturbance, and it is practically the steady displacement 
appropriate to the momentary state of the source, accompanied 
by a weak state of magnetic force connected with the displace- 
ment nearly in the instantaneous manner. That is, the time- 
variation of the displacement is the electric current, and the 
displacement itself is sensibly that of the static theory. Under 
these circumstances there is next to no waste of energy. 

But by increasing the frequency we can completely alter this 
state of things. For we increase the rate of change of the 
impressed force, and therefore the strength of the electromagnetic 
waves generated, for one thing. Besides this, we can bring the 
region of electromagnetic waves nearer to the source, and so 
contract the region in which the displacement was approxi- 
mately static as much as we please. In the limit with very great 
frequency we have nearly pure electromagnetic waves close up 
to the source. The waste of energy increases very rapidly with 
the frequency, varying as its fourth power with simple-harmonic 
waves. The extremest case of waste is that of a discontinuity, 
already mentioned, as when setting up the steady state we 
waste half the work done, and then, when the force is removed, 
waste the other half. 

Self-contained Forced Electromagnetic Vibrations. Contrast 
with Static Problem. 

179. When the sphere of impressed force is of finite size, 
another effect comes into view of a somewhat striking character, 
of which the theory of a point-source gives no information. By 
increasing the frequency sufficiently, we shall reduce the ampli- 


tude of the external vibrations, and on reaching a certain fre- 
quency depending upon the size of the sphere, they will entirely 
vanish. That is, at a certain frequency, a simple harmonic 
impressed force, acting uniformly within a spherical space in a 
dielectric, produces no external effect whatever. The vibrations 
are stationary or standing, and are purely internal, or confined 
to the sphere itself. By general experience of statical problems 
this would seem to be impossible. But we must expect to find 
strange manners when we go into strange lands. 

This strange behaviour arises from the general electromag- 
netic property that the true source of disturbances due to im- 
pressed force (electric or magnetic) is the curl thereof. The 
impressed force itself, on the other hand, is associated with the 
supply of energy. (See 87.) In the present case the sur- 
face of the sphere of impressed force is the seat of its curl, its 
intensity varying as the cosine of the latitude, if the polar axis 
be parallel to the impressed force ; and consequently, as already 
described for the case of setting up the steady state, the elec- 
tromagnetic waves proceed both ways from the surface at 
which they originate. The waves going inward contract, then 
cross at the centre and expand again. We have, therefore, 
two trains of outward waves, and the external effect may be 
imagined to be due to their superimposition. That they may 
sometimes assist and sometimes partly cancel one another, 
according to relative phase, may be readily conceived, and 
the theory indicates that at any one of an infinite series of 
definite frequencies there is complete cancellation externally. 
This refers to the simple-harmonic state. There is always an 
external effect initially, at any frequency namely, the initial 
electromagnetic shell of depth equal to the diameter of the 
sphere of impressed force, because it consists entirely of the 
beginning part of the first outward wave, the second one only 
reaching to its rear ; but on the inner side of this shell there 
may be no disturbance, except within the sphere of impressed 
force. The waste of energy is not continuous at the critical fre- 
quencies, but consists merely of the energy in the initial shell. 
The impressed force works continuously, but as much positively 
as negatively within a period, so that it is inactive on the whole. 

Effects of the above described kind are principally remark- 
able from the contrast they present towards more familiar 


effects, especially those of a statical kind. The behaviour of 
electrification on a sphere may be taken as an illustrative con- 
trast. We know that if we have a spherical surface in a uni- 
form dielectric covered with electrification, the displacement is 
partly internal and partly external. Only when the electri- 
fication is uniformly spread does the displacement vanish 
internally, and become wholly external. This was for- 
merly explained by the inverse-square law. Every element 
of electrification was supposed to exert force equably in 
all directions round itself, internally as well as externally, 
and a mathematical consequence of this is the complete 
cancellation of the internal force when the electrification 
is uniformly spread on the surface of a sphere. The same 
reasoning was applied when the sphere was not of the same 
nature as the external medium when it was a conductor; to 
wit. Now the internal force is also zero in this case. More- 
over, it is zero whatever be the nature of the internal material 
(without electrical sources). But it is certainly untrue that 
the electric force due to a point-source of displacement is the 
same in all directions around it when the medium is not homo- 
geneous. So the reason for the absence of internal force is 
entirely wrong, although it seems right in one case, viz., that 
of homogeneity. The true, and sufficient, and comprehensive 
reason for the equilibrium, and the absence of internal force is 
the satisfaction of the second circuital law when the external 
electric force is perpendicular to the surface. This is independent 
of the nature of the internal matter, and is obviously to be pre- 
ferred to a hypothesis that is only valid sometimes. Varying the 
internal material will make the law of spreading of force be 
different in any number of ways, and, if we like, different 
for every element of electrification, by arranging various 
kinds of heterogeneity. But independently of this, we might 
vary the law of force in many ways in special cases. 
Thus, in the case of a uniform spread of electrification 
on a sphere, we may imagine the displacement from any 
particle of electrification to radiate in any symmetrical manner 
that will give zero displacement inside and a uniformly 
radial displacement outside. The simplest case after the usual 
uniform spreading in all directions, is a planar spreading. Let 
every element of electrification on the surface send out its 


displacement in the tangential plane only, though equally in 
all directions in that plane. This will give the correct static 
result. It is not altogether a fantastic example, for we can 
make an electromagnetic problem of it by letting the electrified 
surface expand at the speed of light. Then the assumed law 
of spread is the actual law, for every element of electrification 
is the core of a plane electromagnetic wave. The resultant mag- 
netic force due to all the waves, however, is zero, and the resultant 
electric force is as in the static problem. (See 61 and 164.) 

Now when we have vibrating electromagnetic sources on 
a spherical surface producing electromagnetic waves simple- 
harmonically, we also in general have both internal and 
external effects, for there is an internal as well as an exter- 
nal train of waves. But whilst they cannot cancel internally, 
they may do so externally. Here is one contrast with the 
static problem, and along with it is another, viz., that it is the 
nature of the external medium that is now indifferent (with a 
reservation), whereas in the static problem we have independence 
of the nature of the internal medium. The reservation hinted at 
is connected with the initial uncancelled electromagnetic shell. 
The external medium should allow it to escape, or absorb it 
somehow, so that it will not interfere with the effects under 
consideration. To regard the zero external effect as being 
actually due to the coexistence of two trains of waves which 
cancel one another is merely a mathematical artifice, however, 
because neither train of waves exists. If they did exist we 
should have to vary their nature to suit the constitution of the 
external medium, just as in the problem of static equilibrium 
we require different laws of force to suit the nature of the 
internal medium. The comparison of the static with the 
kinetic problems bhows a complete reversal of relations as 
regards internal and external. 

What we can do with a single surface electromagnetic source 
we can repeat with others inside it. We see, therefore, that 
the whole sphere of impressed force may be filled up with 
vibratory sources of the most vigorous nature without producing 
any (except initial) external effect, if their periods be properly 
chosen in relation to their situation, which problem admits of 
multiple solutions. This is suggestive as regards the stores o> 
energy bound up with matter. 


Relations between E and H in a Pure Wave. Effect of Self- 
induction. Fatuity of Mr. Preece's " KR law." 

180. In the above very little has been said about the dis- 
tribution of electric and magnetic force from part to part of 
an electromagnetic wave. This varies greatly in different 
kinds of waves, and cannot be explained without the formulae. 
But there is one very important property of pure electromag- 
netic waves (which also holds good, more or less approximately, 
in general) which may be described at present. As already 
mentioned, the electric and magnetic energies are equal in a 
pure electromagnetic wave. Now the density of the electric 
energy is -JcE 2 , if E is the electric force (intensity), and that 
of the magnetic energy is J/*H 2 . If we equate these, we obtain 
a relation between E and H. Thus 


Also /*cv 2 = l ..... (2) 

Therefore E = jj.vR ....... (3) 

The positive or negative sign depends on which way the wave 
is going. Disregarding this, the electric and magnetic forces 
have a constant ratio. They are therefore in the same phase, 
or keep time together in all their variations without lag or 
lead. Along with this, they have the property of being per- 
pendicular to one another (that is, E and H are perpendicular, 
not E and H), and their plane is in the wave-front, or the 
direction of motion of the wave is perpendicular to E and to H. 
It is the direction of the flux of energy. These properties hold 
good in all parts of a pure electromagnetic wave, although the 
magnitudes and directions of the electric and magnetic forces 
may vary greatly from one part of the wave to another. 

The above is also the state of things that obtains, more or 
less perfectly or imperfectly, in long-distance telephony over 
copper circuits of low resistance, and by lowering the resistance 
per mile we may approximate as nearly as we please to the 
state of pure electromagnetic waves. It is the self-induction 
that brings about this state of things, showing such a contrast 
to the more familiar relations between the electric and magnetic 
forces on circuits in general. Like a kind of fly-wheel, the 
self-induction imparts inertia and stability, and keeps the waves 


going. It is the long-distance telephoner's best friend who 
was, not many years since, spurned with contempt from the 
door. Some people thought there was a very absurd fuss made 
about self-induction, and that it was made a sort of fetish of ; 
so, knowing no better, they poured much cold water on the 
idol. But, whatever opinions we may hold regarding their 
competence as judges, there can be no question about the stern 
logic of facts. Self-induction came to stay, and stayed it has, 
and will stay, having great staying power. Whatever should 
we think of engineers who declined to take into account the 
inertia of their machinery? There was also some consider- 
able fuss made about a supposed law of the squares, or KR 
law as it was or is called, according to which you could 
not telephone further than KR = such or such a number, 
because the speed of the current varied as the square of 
the length of the line, or else inversely. But in spite of the 
repeated attempts made to bolster up the KR law, the critical 
number has kept on steadily rising ever since. I see now that 
it has gone up to 32,000.* But it need not stop there. Make 
your circuits longer, and it will go up a lot more. 

As regards the ether, it is useless to sneer at it at this time 
of day. What substitute for it are we to have ? Its principal 
fault is that it is mysterious. That is because we know so 
little about it. Then we should find out more. That can- 
not be done by ignoring it. The properties of air, so far 
as they are known, had to be found out before they became 

Wave-Fronts; their Initiation and Progress. 

. 181. Still keeping to a simple dielectric, we may always, 
by consideration of the fact that the speed of propagation is v, 
find the form of the wave-front due to any collection of point- 
sources, and trace the changes of shape and position it under- 

* See The Electrician, December 30, 1892, p. 251, for data, in article by 
Mr. Jos. Wetzler. The 32,000 is for the New York-Chicago circuit of 1,000 
miles at 4'12 ohms per mile, or 2'06 ohms per mile of wire. Comparison of 
Mr. Wetzler's with Mr. Preece's figures is interesting. Good telephony is 
got by Mr. P. at 10,000, and by Mr. W. at 45,000 ; excellent by Mr. P. at 
5,000, and by Mr. W. at 31,000 ; and so on. But there is still less of a 
KR law in the American than in the English cases. 



goes as It progresses. For obviously, if a point P be at a dis- 
tance vt from the nearest source, no disturbance can have 
reached P from it if it started into action after the moment 
t = ; and generally, the effect at P at a given moment arising 
from a particular source depends upon its state at the moment 
r/v earlier (if r be the distance from the source to P), and upon 
the previous state of the source, causing residual or cumula- 
tive action at P. We therefore know the limiting distance of 
action of the sources for every one of their momentary states. 
The wave-front belonging to a set of disconnected point-sources 
consists initially of disconnected spheres. But as they expand, 
they merge into one another to form a continuous extreme wave- 
front consisting of portions of spheres. When the point-sources 
are spread continuously over a surface to form a surf ace-source, we 
have a continuous wave-front from the first moment ; or rather, 
two wave-fronts, one on each side of the surface. There is an 
exception, to be mentioned later, when the sources send waves 
one way only from the surface. This is not the already 
described case of the cancelling of two trains of waves at par- 
ticular frequencies, but is a unilateral action obtained by a 
special arrangement of surface-sources. Passing over this, 
observe that when we have once got a wave-front we may ignore 
the sources which produced it, and make the wave-front itself 
tell us what its subsequent history will be. For, to trace the 
course of the wave-front from one moment to the next, we have 
merely to move any element of the surface in the direction of 
the normal to that element through a small distance a to 
obtain the new position at the moment a/v later. This being 
done for the whole surface, gives us the new position of the 
wave-front, and by continuing this process we may follow the 
wave-front in its progress as long as we please. Thus the wave- 
front coming from the surface of a sphere is a sphere ; from a 
round cylinder, if infinitely long, also a round cylinder ; but if 
of finite length, then a cylinder with rounded ends ; from a cube, 
initially a cube with rounded corners, but becoming more and 
more spherical as it expands ; and so on. It is easily seen that 
at a sufficiently great distance from a finite collection of sources 
of any kind, the wave-front tends to become spherical, or the 
complex source tends to become equivalent, at a great distance, 
to a point-source of some complex kind. 


Effect of a Non-Conducting Obstacle on Waves. Also of a 
Heterogeneous Medium. 

182. Now consider the effect of an obstacle brought into 
our medium, say a non-conducting dielectric mass of different 
inductivity or permittivity. The manner of propagation in it 
is similar in kind to that in the external medium, bat it varies 
in detail, and the speed will usually be different. Then, when 
we have a wave coming from (say) a point- source in the first 
medium, the presence of the obstacle at first makes no dif- 
ference. It has no immediate action, and, until the wave 
reaches it, might as well not be there. But immediately the 
wave does reach it a change occurs. The interface of the two 
media becomes the seat of sources of fresh disturbances, or they 
may be considered fictitious sources, in contrast with the 
original, for they do not bring in any fresh energy. Thence 
arises a new effect, viz., reflection. Not the whole, but only a 
part of the wave disturbance enters the new medium. The rest 
is thrown back, and forms a reflected wave. In the same way 
as we may trace the course of primary waves from a source in 
a uniform medium may we trace the course of the secondary 
waves set up by the obstacle. The disturbance outside it is 
then due to the superposition of the primary and secondary 
waves, and, if there be just one obstacle, this state of things 
continues. Of course the secondary wave- front may itself be 
complex, because the disturbances going into the second me- 
dium and transmitted therein according to its nature, may 
reach the interface again at other parts, and there suffer re- 
flection and transmission anew. This complication is done 
away with by making the obstacle infinitely big, with a plane 
boundary. Then we have just one wave in the obstacle and 
two in the medium containing the source, viz., the primary and 
the first (and only) reflected wave. 

But if there be a second obstacle, not only will it, like the 
first, give rise to a secondary wave when the primary meets it, 
but each obstacle will act similarly towards the secondary wave 
from the other, whence arises a pair of tertiary waves, and so 
on. Thus, with only two obstacles, we shall have a succession 
of infinitely numerous waves crossing and recrossing one 
another, arising out of the action of a point-source, with 



great complications. But, however complex in detail, we have 
the important fact that the mere knowledge of the speed of 
propagation allows us to lay down the whole course of the 
waves generated, and the position of the wave-fronts belonging 
to a given epoch at the source. The mathematics of the full 
treatment may be altogether beyond human power in a reason- 
able time; nevertheless, we can always predict with confidence 
that the results must have such and such general properties 
relating to the course of the waves, besides the properties 
involved in the persistence of energy. 

In the above we were concerned with discontinuous he- 
terogeneity, or an abrupt change in the value of one or both 
of the constants c and /* at an interface. When the medium is 
continuously heterogeneous, or the "constants" change in 
value continuously from place to place, then the speed v is 
made a function of position. The process of partial reflection 
and partial transmission which occurred at the interface now 
takes place in general wherever the value of c or p changes. 
These changes being continuous, so are the results, so that we 
do not have distinctly separable trains of waves, but rather a 
continuous distortion. We can, however, follow the course of 
a wave expanding from a point by communicating to the wave- 
front the speed proper to its position, such speed being, as 
above, definitively known. A spherical wave may remain 
spherical, only varying in its speed; or more likely, it may 
change its form as it expands. It may become ellipsoidal, for 
example, and of course must do so if the speed in different 
parts of the medium vary suitably. 

It is possible for the inductivity and permittivity to change, 
either abruptly or continuously, without any reflection, or cast- 
ing behind of the disturbance as a wave progresses, although 
its speed varies. In such a case it will be found that the ratio 
IJL/C is constant. The change in p,, or the ratio of induction to 
magnetic force, then fully compensates the change in c, the 
ratio of displacement to electric force, either of which changes 
by itself would cause reflection. 

In connection with a heterogeneous medium (including abrupt 
changes of nature), we may notice its behaviour as regards 
steady states, in contrast with that of a thoroughly homo- 
geneous medium. In the latter case, as we have already 


described, the introduction of a steady point-source causes 
the steady state to begin immediately at and spread around it, 
after which no change occurs. That is, the medium being 
homogeneous, there is no reflex action. But when it is hetero- 
geneous the case is quite different. The steady state of dis- 
placement due to impressed voltage depends upon the per- 
mittivity in all parts of the medium, and only comes about as 
the final result of the infinitely numerous reactions between 
different parts, due to the reflections that take place, which 
modify the final distribution of displacement as well as (usually) 
the total amount. There is a similar contrast in the mechanism 
of the mathematics involved in the calculation of the steady 
state. When the medium is homogeneous we may find it 
through the potential of fictitious matter at the source alone. 
When heterogeneous, we require to have fictitious matter all 
over the medium, or rather in all parts where it changes its 
nature, and, owing to the infinitely numerous reactions, we 
should, in a general or complete solution, require to perform 
not one space-integration, but an infinite series of successive 
space-integrations, before we could arrive at the true potential 
function, allowing for the variations of permittivity every- 
where. Thus, by electromagnetic considerations we obtain 
some insight of the true nature of transcendental static prob- 
lems, involving potential functions and assumed instantaneous 
action at a distance. 

Effect of Eolotropy. Optical Wave -Surfaces. 
Electromagnetic versus Elastic Solid Theories. 

183. When the medium is not isotropic as regards either 
the displacement or the induction, or as regards both, we have 
very remarkable effects, known in the science of optics as 
double-refraction. There can be two distinct wave-speeds for a 
given position of the wave-front, and these speeds change when 
the direction of the normal to the wave-front changes. Hence 
double refraction, or the separation of a wave entering an 
eolotropic medium into two waves travelling independently of 
one another at different speeds. If the medium is only elec- 
trically eolotropic, although the displacement and induction 
are still in a wave-front (of either wave), and are perpendicular 
to one another, the electric force is inclined to the wave-front, 


so that the ray, which indicates the direction of the flux of 
energy, is inclined to the normal to the wave-front. Similarly, 
with magnetic eolotropy alone, it is the magnetic force that is 
inclined to the wave front. When the medium is both elec- 
trically and magnetically eolotropic, both the electric and the 
magnetic force are inclined to the wave-front, whilst the dis- 
placement and induction, though still in the wave-front, are no 
longer perpendicular to one another. 

By imagining plane electromagnetic sheets to traverse the 
medium in all possible directions about a point, and comparing 
their positions with respect to the point at equal times after 
crossing it, we arrive at the conception of the wave-surface. 
This is obviously a sphere in an isotropic medium. But when 
it is eolotropic, the two speeds, and their variation according 
to the direction of motion of the waves, make the wave-surface 
become a very singular double surface. With electric eolotropy 
alone, which is the practical case, it is Fresnel's wave-surface. 
It is also another Fresnel surface with magnetic eolotropy 
alone. But when the medium is eolotropic as regards both the 
induction and the displacement, the wave-surface is of a more 
general and symmetrical character, including the former two 
as extreme examples. It is still a double surface, however, 
except in one case. We have already mentioned that in an 
isotropic medium there is a peculiar behaviour when the ratio 
P./C is constant, although fj. and c vary. We might anticipate 
some peculiarity in the wave-surface when /z/c is constant. 
This constancy now means that the directional properties of //, 
are exactly paralleled by those of c. That is, the principal 
axes of p and those of c are coincident, whilst the value of the 
ratio of the permittivity to the inductivity is the same for the 
three axes. The result is to reduce the double wave-surface to 
a single surface, which is an ellipsoid. 

Assuming light to consist of vibrations of an " elastic solid " 
medium, single or ordinary refraction is explained by an 
alteration in the density or the rigidity of the ether. Not 
only is the theory quite hypothetical in considering the kinetic 
energy to be the energy of vibrational motions of displacement, 
but the alteration of density or rigidity assumed is a further 
hypothesis on the top of the main one, for there is no evidence 
that there is such a change. Moreover, the explanation will 


not work properly. On the other hand, the electromagnetic 
theory says that light consists of electromagnetic vibrations in 
the ether. This, too, is a hypothesis. But the auxiliary part, 
that refraction is caused by change of permittivity from one 
medium to another, is not a hypothesis, but a fact. More- 
over, the theory works. 

Similarly, double refraction in elastic solid theories of light 
is explained by eolotropy as regards elasticity, or by something 
similar relating to the density. This is also hypothetical, and 
not without its troubles. But, on the other hand, Maxwell 
declared that double refraction occurs because the doubly 
refracting medium is electrically eolotropic. Now this is a 
fact too, and the theory is a clear one. 

These remarks will serve to illustrate what I mean by the 
far greater intrinsic probability of the electromagnetic theory, 
apart from the experimental work of late years. It is much 
less hypothetical than elastic solid theories. We know nothing 
about the density or the rigidity of the ether, or how they vary 
in different bodies, or' if they vary. But we do know a good 
deal about electric permittivity and magnetic inductivity. In- 
stead of dealing with possibilities, we are dealing with actual facts. 

The old objection that a mechanical theory of light was 
surely to be preferred to an abstract electromagnetic theory 
was very misleading. The electromagnetic theory is mechan- 
ical, without, however, a precise specification of the mechanism. 
An elastic solid theory is merely a special mechanical theory. 
It cannot satisfy the electromagnetic requirements, but this 
failure, though immensely important in itself, is not the point 
here. Even if it did satisfy them, it would probably be less 
true than the electromagnetic theory, which, being abstract, 
does not assert so much. There may be many "mechanical" 
solutions of an abstract theory. Elastic solid theories are a 
great deal too precise in saying what light consists of, and 
mechanical speculations in general should be received with 
much caution, and regarded rather as illustrations or analogies 
than expressions of fact. We do not know enough yet about 
the ether for dogmatising. 

One can imagine that a clear-headed man might be able to 
work his way through all the theories of light yet propounded, 
assimilate what was useful and true, eliminate what was useless 


or false, and finally construct a purified theory, not professing 
to explain what light is, but still connecting together in a per- 
fectly unobjectionable manner, free from hypotheses, all the 
principal facts, and of so compliant a nature as to readily adapt 
itself to future discoveries, and finally settle down to a special 
form expressing the theory of light. It is hardly likely that 
the clear-headed man will be found for the purpose, and per- 
haps his theory would not be very popular. It would be so 
very abstract. What is far more likely is that the electro- 
magnetic theory (itself abstract and possessing many of the 
desired qualifications), which has already begun to find its 
way into optical treatises at the end, will gradually work its 
way right through them to the beginning, and in doing so, 
oust out the most of the old-fashioned hypotheses. The 
result will be to have optical theory expressed throughout in 
electromagnetic language. To do it properly, it is hardly 
necessary to say that the preposterous 47r, which the B.A. 
Committee seem to want to perpetuate, should be ignored from 
the beginning. 

A Perfect Conductor is a Perfect Obstructor, but does not 
absorb the Energy of Electromagnetic Waves. 

184. So far, in varying the nature of the medium, we have 
not introduced any property causing its local waste, such as 
the existence of conductivity (finite) necessitates. In all the 
varied journeys of electromagnetic waves in a (theoretical) non- 
conducting dielectric with non-conducting obstacles there is no 
local waste of energy, and the work done by impressed forces is 
entirely accounted for by the electric and magnetic energy. 
Assuming a certain amount of work done up to a certain 
moment, and none later, that amount expresses the energy of 
the electromagnetic field, and, however it may vary in distribu- 
tion, the total remains the same, just as if it were a quantity 
of matter moving about, having continuity of existence in time 
and space. The useless complication introduced by the cir- 
cuital indeterminateness of the energy flux may be ignored. 
The only way, then, to get rid of the energy is to absorb it at 
the sources by working against impressed forces. Putting 
that on one side, there is the waste of energy by dissipation 
in space, which cannot be stopped. The energy is still in exis- 


tence, and is still in the electric and magnetic forms, and must 
always be at a finite distance from its source and an infinite 
distance from infinity ; but since it gets out of practical reach, 
and is constantly going out further, it is virtually wasted. 
Thus, ultimately, the energy of all disturbances generated in 
a boundless non-conducting dielectric goes out of range and is 

To prevent this, we may interpose a reflecting barrier. 
Imagine a screen to be introduced, at any finite distance, 
enveloping the sources, of such a nature as to be incapable of 
either transmitting or absorbing the energy of waves impinging 
upon it. Then clearly the dissipation of the energy is stopped, 
and it all keeps within the bounded region. The waves from 
the source will be reflected from the boundary, and their sub- 
sequent history will be an endless series of crossings and re- 
crossings. The only way to destroy the induction and dis- 
placement is to employ artful demons (or impressed forces), so 
situated and so timing themselves as to absorb the energy of 
waves passing them, instead of generating more disturbances. 
In the absence of this demoniacal possession of the region, the 
energy will remain within it in the electromagnetic form and 
be in constant motion. 

But any opening in the screen, establishing a connection 
through ether between the inner and outer 'regions, will at 
once put a stop to this local persistence of energy. For energy 
will pass through the opening, and once through, cannot get 
back again (though a part may), but will escape to infinity. A 
mere pinhole will be sufficient, if time enough be given, to allow 
all the energy to pass through it into the external region, and 
there go out unimpeded to an infinite distance, in the absence 
of a fresh complete screen to keep it within bounds. 

A screen to perform the above-described functions may be 
made of that very useful scientific substance, the perfect con- 
ductor, which is possibly existent in fact at the limiting zero 
of absolute temperature, the latest evidence being the recent 
experiments of Profs. Dewar and Fleming, measuring the resist- 
ances of metals at very low temperatures. If a perfect con- 
ductor, it is also a perfect obstructor, or is perfectly opaque to 
electromagnetic waves, without, however, absorbing their 
energy superficially. 


Conductors at Low Temperatures. 

185. For this reason, a perfectly conducting mass exposed 
to electromagnetic radiation in ether cannot be heated thereby, 
for heating would imply the absorption of energy. If, then, 
the law (whose existence has long been suspected) that metals 
tend to become perfect conductors when their temperature is 
sufficiently lowered, were absolutely true, it would follow that 
a metal, if once brought to zero temperature, would remain 
there, provided its only source of heat-energy be electromag 
netic vibrations. At the same time it is conceivable, and, in- 
deed, inevitable, that it should receive energy from them to 
some extent in another way, viz., by the electromagnetic stress 
producing motion in bulk, even though the establishment of 
irregular molecular motions be prevented. 

Similarly, we should expect that a metal which obeys the law 
approximately would show very small absorptive power for 
radiation at very low temperatures. This refers to the recep- 
tion of the energy of true radiation in ether that is, in vacuo. 
A body may become heated in other ways, by the impacts of 
air particles, for example, if air be present. Prof. De war's late 
experiments are suggestive in this respect, but it is too soon 
to draw conclusions from partially-published experiments. 

But even as regards strongly absorptive bodies, the perfectly 
black body of the thermal philosophers, for example, we may 
expect a similar diminution of absorptive and emissive power 
with fall of temperature. Stefan's law of radiation asserts that 
the emissivity varies as the fourth power of the absolute tem- 
perature. Since, however, in deriving this law from electro- 
magnetic principles, as has been recently done by B. Galitzine,* 
we seemingly require to invoke the aid of reversible cycles and 
the second law of thermodynamics, whose range of application 
is sometimes open to question, we may well be excused from 
an overhasty acceptance of the law as the expression of abso- 
lute fact. The second law of thermodynamics itself needs to 
be established from electromagnetic principles, assisted by the 
laws of averages, so that we may come to see more clearly the 
validity of its application, and obtain more distinct notions of 
the inner meaning of temperature. 

* Phil. Mag., February, 1893. 


Equilibrium of Radiation. The Mean Flux of Energy. 

186. As before said, a perfectly conducting screen enclos- 
ing a dielectric region supporting electromagnetic disturbances, 
keeps in their energy, which remains in the electric and mag- 
netic forms, and if there be no sources of energy present, the 
total energy remains constant. Some interesting questions 
arise regarding the subsequent history of the electromagnetic 
disturbances, when left to themselves, subject to the obstruct- 
ing and reflecting action of the screen. 

In certain cases the initial state will continue absolutely 
unchanged ; for example, when it is the steady state due to 
electrification, or associated therewith. Similarly as regards 
an initially steady state of magnetic force that, for instance, 
associated with a linear current (without resistance) in the 
enclosed region. Again, in other cases, although the subse- 
quent history of an initial state may be one of constant 
change, yet there will be regular recurrence of a series of 
states ; as when a periodic state of vibration persists without 
any tendency to degrade. It is sufficient to mention the very 
rudimentary case of a plane wave running to and fro between 
parallel plane reflecting boundaries, without the slightest ten- 
dency to change the type of the vibrations. We see from 
these examples, which may be multiplied, that there is no 
necessary tendency for the initial state, even when vibratory, 
to break up and fritter down into irregular vibrations. Never- 
theless, there does appear to be a general tendency to this 
effect, when the initial states are not so artfully selected as to 
prevent it happening. Even when we start with some quite 
simple type of electromagnetic disturbance, the general effect 
of the repeated reflections from the boundary (especially when 
of irregular form) and the crossing of waves is to convert the 
initial simplicity into a highly complex and irregular state of 
vibration throughout the whole region. This cannot happen 
universally, as we have seen, and therefore a general proof of 
conversion from any initial state to irregularity cannot be 
given; but there can be little doubt as to the usual possi- 
bility of the phenomenon. Especially will this be the case 
if the initial state be itself of an irregular type, such as that 
due to ordinary radiation from matter, when it is tolerably 


clear that the irregularity will persist, and become more 

Let us, then, assume that we have got the medium within our 
screen into this state in its extreme form, without enquiry into 
the intermediate stages. Then the very irregularity gives rise 
to a regularity of a new kind, the regularity of averages. The 
total energy, which is a constant quantity, will be half electric 
and half magnetic, and will be uniformly spread throughout 
the enclosure, so that the energy density (or energy per unit 
volume) is constant. As regards the displacement and the 
induction, they take all directions in turn at any one spot, quite 
irregularly, but so that their time-averages show no directional 
preference. Similarly, the flux of energy, which is a definite 
quantity at a given moment at a fixed spot, is constantly 
changing in amount and direction. But in virtue of the con- 
stancy of the mean density and the preservation of the normal 
state by constant exchanges of energy, there is a definite mean 
energy flux to be obtained by averaging results. This mean 
flux expresses the flux of energy per second across a unit area 
anywhere situated within the enclosure. 

To estimate its amount, let the mean density of the energy 
be U. This is to include both kinds. Now fix attention upon 
a unit area, A, fixed in position anywhere within the enclosure, 
and consider the flux of energy through it under different cir- 
cumstances. First of all, if the energy all moved the same 
way at the same speed v (that of propagation), as in simple 
plane progressive waves, and the direction of its motion were 
perpendicular to the fixed unit area A, then the energy passing 
through it would belong to a ray (or bundle of rays) of unit 
section, and the energy flux would be Uv simply. This is the 
maximum. But this is impossible, because energy would accu- 
mulate on one side of A at the expense of the other. The 
next approximation, to prevent the accumulation, is to let half 
the energy go one way and half the other ; still, however, in 
the same line. This brings us down to JUv. To go further, 
we must take all possible directions of motion into account. 
The original ray conveying Uv must assume all directions in 
turn, and the mean value of the flux through A must be 
reckoned. Now, if the ray of unit section be turned round 
so as to make an angle with the normal to A, the effective 


section of the ray is reduced from 1 to cos ; that is, cos 6 is 
the fractional part of the ray which sends its energy through 
A, through which the flux per second is therefore Uv cos 0, due 
to the ray at inclination 6. The true flux through the area A 
is therefore the mean value of Uv cos for all directions in space 
assumed by the ray. Now the mean value of cos for a com- 
plete sphere is zero, and therefore the mean flux through A is 
zero. This is right, as it asserts that as much goes through 
one way as the other. To obtain the amount going either way 
we must average over a hemisphere only. The mean value of 
cos 6 is then J. But we are only concerned with half the total 
energy, or JU, when we are confined to one hemisphere. Con- 
sequently we have 

W = JU*, ..... . (4) 

to express the flux of energy W per second each way through 
any unit area in the enclosure. 

Another way of getting this result, which is, however, essen- 
tially the same, is to divide the original ray of unit section 
along which the flux is Uv into a very great number n of equal 
rays of unit section, each conveying \/n part of the same, and 
placed at such inclinations to the normal to A that no direction 
in space is favoured. This amounts to dividing the surface of 
a sphere whose centre is that of the area A into n equal parts, 
the centre of every one of which defines the position of one of 
the n rays. Any ray now sends (Uv/n) cos through A per 
second. Now sum this up over the whole hemisphere and the 
result is W. In the limit, when n is infinitely great, we have 

e e=i 


as before. The 4:r divisor in the integral is not the unspeak 
able 4?r of the B.A. units. It is the area of the sphere of unit 
radius, whilst the other factor sin ddd<j> is the area of an 
element of the sphere. 

As this is an important fundamental result in radiation, it 
is desirable to establish it as generally and simply, and with as 
much definiteness of meaning as possible. Bartoli made it 
come to JU0 (apart from electromagnetic considerations, which 


are, however, only accidentally involved). On the other hand, 
Galitzine makes it JUfl by considering a special case, viz., 
that of a cylindrical enclosure with flat ends, one of which is a 
radiant source (perfectly black) at constant temperature, whilst 
the other end and the round surface are perfectly reflecting. 
It would appear, however, from the above method, that the 
result is general, and is independent of sources of heat, and of 
the emissivity and temperature. Since we made no use of the 
screen after introducing it to keep in the radiation, it may be 
dispensed with, provided the stationary condition be still main- 
tained. Thus, if a portion of the screen be made " perfectly 
black," maintained at constant temperature, the quantity W, 
which represents the amount of heat falling upon it per unit 
area per second, is completely absorbed by it. But this is 
perfectly compensated by the emission from the black surface 
of an equal amount of heat. So %Uv measures the total emis- 
sivity under the circumstances. 

The Mean Pressure of Radiation. 

187. Another important fundamental quantity is the mean 
pressure of radiation. In a simple ray the electric and mag- 
netic stresses unite to form a pressure U along the ray, with 
no pressure or tension in lines perpendicular to the ray, that 
is, in the plane of the wave, as described in 86. But when 
the radiation is balanced as in the last paragraph, there is no 
directional preference, and the pressure is all ways in turn, and 
therefore, on the average, simulates a hydrostatic pressure. Its 
value may be readily estimated in a manner similar to that 
employed above concerning the mean flux of energy. When 
we make the energy U go all one way in a ray of unit section 
through the area A situated anywhere, the pressure in the ray 
is U, and this is the pressure on A if the ray is perpendicular 
to A. But when the ray is inclined to the normal to A at an 
angle 6, only the fraction cos of the ray is concerned in the 
action upon A. Furthermore, the line of pressure is inclined 
to the normal to A at an angle 0, so that the effective normal 
pressure is still further reduced by the factor cos 6 a second 
time. The pressure on A is therefore only Ucos 2 0. This 
must now be averaged for all directions in space that we may 
give to the ray, without preference. Now the mean value of 


cos 2 for the complete sphere is J. So the pressure, say p, is 
given by 

*> = JU ..... .... (6) 

Notice particularly that we take the mean for the complete 
sphere, not merely for the hemisphere as in the former calcula- 
tion. The reason is that whether a ray goes through A from 
right to left or from left to right, the pressure is the same : so 
both ways have to be reckoned. 

Or, we may divide the original ray in which the pressure is 
U into a great number n of rays also of unit section, in each of 
which the pressure is \J/n. Let the axes of these rays be denned 
by the middle points of n small equal areas into which we may 
divide the surface of a sphere, and sum up the normal pres- 
sures on A. We obtain, in the limit, 

This result was given by Boltzmann some years ago, and 
Galitzine confirms it for the special cae of his straight cylinder 
with a radiant surface at one end. By the above method we 
see that the result is general, resulting from the uniformity 
of radiation in all directions, as the previous formula for the 
mean flux of energy did. 

Emissivity and Temperature. 

183. This pressure p is not only the mean pressure 
throughout the enclosure, but also the pressure on its enve- 
lope, which exerts an equal back pressure. If it move, then 
work is done by or against the enclosed radiation through 
the agency of its pressure, and the enclosure loses or gains 
energy to an equal extent. Observe that the idea of tempera- 
ture does not enter explicitly when the boundary of the enclo- 
sure is of ideal perfectly reflecting material. But when it, 
or a part of it, is made absorptive and emissive, the physics 
of the matter becomes far more difficult and to some extent 
dubious. The notion of temperature comes in, and with it the 
isecond law of thermodynamics. Assuming its full applica- 
bility, Stefan's law follows easily enough, as Galitzine has 
shown. Let the enclosure bo a cylinder of unit section with 
two pistons A and B. Let A be fixed, and be a perfectly black 


body, whilst B Is movable to and from A and is a perfect 
reflector, as is also the round cylinder. 

Now start with B close to A ; the volume of the enclosure 
is then nil. Draw B very slowly away from A through the 
distance h t keeping A at constant temperature t all the time, 
and then stop it. During this operation the source A keeps 
the enclosure filled with energy to density U, and pressure p, 
corresponding to the temperature t at which A is maintained. 
The pressure p does the external work ph. Besides that, there 
is energy UA in the enclosure at the end of the operation. 
Their sum is therefore the heat lost by A (excess of heat 
emitted by A into the enclosure over that returned to A). 

H-(U+i>)A (8) 

Now we know p in terms of U, so that we have, by (6), 
H = |TO (9) 

Now, B being fixed, let the cylinder cool down to zero tempera- 
ture. The whole of the energy UA in the enclosure goes out 
through A. Lastly, push B back to A without working, and 
then raise A to temperature t. A cycle is then completed. 
Applying the second law, we have 


where H is as before, and dR is an element of the heat lost in 
the cooling process. On the left side put (U+p)h or Uh 
for H, because external work was done in the first operation. 
On the right side leave out the p term, because during the 
cooling B was fixed. So we get 

by omitting the factor h. Differentiate with respect to t, and 
we get 

5?-*-* (12) 

dt t ' 

from which we conclude that U, and, therefore, also p and W, 
vary as the fourth power of the temperature, the result above 
mentioned as applied to the emissivity W. 


We tacitly assume that the ether is able to escape freely from 
the cylinder through its envelope, or else that it is freely com- 
pressible, without resistance. This difficulty in connexion with 
the ether is a very old one. 

Internal Obstruction and Superficial Conduction. 

189. The properties of a perfect conductor ara derived from 
those of common conductors by examining what would happen 
if the resistivity were continuously reduced, and ultimately 
became zero. In this way we find that a perfect conductor 
is a perfect obstructor, for one thing, which idea is singu- 
larly at variance with popular notions regarding conduc- 
tors. But it is also a perfect conductor literally, though 
in a different sense to that commonly understood. Ohm's- 
law has played so important a part in the development of 
electrical knowledge, especially on the practical side, that it 
is really not at all a matter of wonder that some practicians- 
should have been so reluctant to take in the idea of a con- 
ductor as an obstructor. Scientific men who can follow the 
reasoning by which the functions of conductors follow from 
known facts have no difficulty in pursuing the consequences- 
far beyond experimental observation. Again, younger men,, 
with fewer prejudices to surmount, do not find much trouble- 
with superficial conduction and internal obstruction. But the- 
old established practitioner with prejudices, who could not 
see the reason, was put into a position of some difficulty 
resembling chancery. If you have got anything new, i 
substance or in method, and want to propagate it rapidly,, 
you need not expect anything but hindrance from the old 
practitioner even though he sat at the feet of Faraday. 
Beetles could do that. Besides, the old practitioner is apt 
to measure the value of science by the number of .dollars- 
he thinks it is likely to bring into his pocket, and if he 
does not see the dollars, he is very disinclined to disturb his 
ancient prejudices. But only give him plenty of rope, and 
when the new views have become fashionably current, he may~ 
find it worth his while to adopt them, though, perhaps, i 
a somewhat sneaking manner, not unmixed with bluster, and 
make believe he knew all about it when he was a little boy t 
He sees a prospect of dollars in the distance, that is the- 


reason. The perfect obstruction having fciiled, try the per- 
fect conduction. 

You should make your converts out of the rising generation 
and the coming men. Thus, passing to another matter, Prof. 
Tait says he cannot understand my vectors, though he can 
understand much harder things. But men who have no quater- 
nionic prejudices can understand them, and do. Younger 
men are born into the world with more advanced ideas, on 
the average. There cannot be a doubt about it. If you had 
taught the Calculus to the ancient Britons you would not 
have found a man to take it in amongst the whole lot, Druids 
and all. Consider too, what a trouble scientific men used 
to have with the principle of the persistence of energy. They 
could not see it. But everybody sees it now. The important 
thing is to begin early, and train up the young stick as you 
want it to grow. Now with Quaternions it is different. 
You may put off till to-morrow what you cannot do to-day, 
for fear you commence the study too soon. Of course, I" 
refer to the Hamilton-Tait system, where you have to do 
violence to reason by making believe that a vector is a 
quaternion, and that its square is negative. 

According to Ohm's law alone, a perfect conductor should be 
one which carried an infinite current under a finite voltage, 
and the current would flow all through it because it does so 
ordinarily. But what is left out of consideration here is the 
manner in which the assumed steady state is established. If 
we take this into account, we find that there is no steady state 
when the resistance is zero, for the variable period is infinitely 
prolonged, and Ohm's law is therefore out of it, so far as the 
usual application goes. In a circuit of no resistance containing 
a finite steady impressed voltage E, the current would mount 
up infinitely and never stop mounting up. On the other hand, 
if we insert a resistance R in the former circuit of no resistance, 
there will be a settling to a steady state, for the current in the 
circuit will tend to the value E/R, in full obedience to Ohm's 
law. The current is the same all round the circuit, although 
a part thereof has no resistance. We conclude that that 
portion has also no voltage. 

But this is only a part of the story. Although we harmonise 
with Ohm's law, we overlook the most interesting part. The 


smaller the resistance the greater the time taken for the 
current to get into the conductor from its boundary, where it 
is initiated. In the limit, with no resistance, it never gets in 
at all. Where, then, is the current? For, as we have said, 
it mounts up to a finite value if there be a finite resistance 
inserted along with the perfect conductor, and mounts up 
infinitely if there be no resistance. 

We recognise the existence of electric current in a wire by 
the magnetic force round it, and in fact measure the current 
by its magnetic force. Therefore, according to this, there is 
the same total current in the wire, if the magnetic force out- 
side it remains the same. If, then, the magnetic force stops 
completely at the surface of the wire, whose interior is entirely 
free from magnetic force, the measure of the current is just 
the same. The uniformly distributed current of the steady 
state appropriate to finite conductivity becomes a mere surface 
current when the conductivity is infinite. In one case we 
have a finite volume-density of current, and in the other a 
finite surface-density. When the current inside the wire is 
zero so is the electric force, in accordance with Ohm's law 
again. The electric and magnetic phenomena are entirely in 
the dielectric outside the wire, the entrance of any similar 
manifestations into it being perfectly obstructed by the absence 
of resistance. For this purpose the thinnest skin would serve 
equally well. In the usual sense that an electric current is a 
phenomenon of matter, it has become quite an abstraction, for 
there is no matter concerned in it It is shut out completely. 
In the circuit of finite resistance, a portion of which is a wire 
of no resistance, supporting a steady current, there is no 
difference whatever in the external, magnetic force outside the 
resisting and non-resisting parts, though in one case there is 
entrance of the magnetic force and waste of energy, whilst in 
the other there is no entrance and no waste. These con- 
clusions do not rest upon Maxwell's theory of dielectrics, but 
upon the second circuital law of electromagnetism applied to 
conductors. But it is only by means of Maxwell's theory that 
we can come to a proper understanding and explanation of the 
functions of conductors. 

The sense in which a perfect conductor is a perfect con- 
ductor in reality as well as in name is that it allows electro- 



magnetic waves to slip along its surface in a perfectly free- 
manner, without waste of energy. Though perfectly obstruc- 
tive internally, it is perfectly conductive superficially. It 
merely guides the waves, and in this less technical sense of 
conduction the idea of a perfect conductor acquires fresh life. 

The Effect of a Perfect Conductor on External Disturb- 
ances. Reflection and Conduction of Waves. 

190. The conditions at the interface of a perfect conductor 
and a dielectric are that the electric force in the dielectric has- 
no tangential component and the magnetic induction no normal 
component. Or 

VNE = 0, NH = 0, 

if N be the unit normal from the conductor. Thus, when 
there is electric force at the boundary it is entirely normal, 
with electrification to match ; and if there is magnetic force it 
is entirely tangential, with electric current to match. Both 
electrification and current are superficial. The displacement 
measures the surface density a- of the one, and the magnetic* 
force that of the other, say c, thus 

<r = ND, c = VNH, 

in rational units, without any useless and arbitrary 4?r constant,, 
such as is required in the B.A. system of units, of amazing 
irrationality. If, then, we have electromagnetic disturbances 
given in a dielectric containing a perfect conductor, the latter 
first of all is free from disturbance, and next causes such re- 
flected waves as to annihilate the tangentiality of the electric 
force and the normality of the magnetic force. 

As regards steady states, the influence of a perfect conductor 
on induction due to foreign sources is to exclude it in the same 
manner as if the inductivity were made zero ; that is, the induc- 
tion goes round it tangentially instead of entering it. This 
is usually ascribed to an electric current-sheet induced upon it& 
surface, whose internal magnetic force is the negative of that 
due to the external field. This is right mathematically, but is 
deceptive and delusive physically. There is no internal force,, 
neither that of the external field nor that of the superficial 
current. The current sheet itself merely means the abrupt 


stoppage of the magnetic field, and cannot really be supposed 
to be the source of magnetic force in a body which cannot 
permit its entrance. The previously mentioned case of a per- 
fectly conducting wire inserted in a circuit of finite resistance 
supporting a steady current, will serve to bring out this point 
strongly. The supposed induced superficial current is now 
actually the main current in the circuit itself. 

It is different with the steady state due to external electric 
sources. The displacement is just as much shut out from the 
perfect conductor (which may also be a dielectric) as was the 
magnetic induction, but in a strikingly different manner, ter- 
minating upon it perpendicularly, as if it entered it in the 
manner that would happen were the conductor nonconducting, 
but of exceedingly great permittivity, so that it drew in the 
tubes of displacement. 

Although a perfect magnetic conductor is, in the absence of 
knowledge even of a finite degree of magnetic conductivity, a 
very far-fetched idea, yet it is useful in electromagnetic theory 
to contrast with the perfect electric conductor. A perfect 
magnetic conductor behaves towards displacement just as a 
perfect electric conductor does towards induction ; that is, the 
displacement goes round it tangentially. It also behaves 
towards induction as a perfect electric conductor does towards 
displacement ; that is, the induction meets it perpendicularly, 
as if it possessed exceedingly great inductivity, without 
magnetic conductivity. This magnetic conductor is also per- 
fectly obstructive internally, and is a perfect reflector, though 
not quite in the same way as electric conductors. The tan- 
gential magnetic force and the normal electric force are zero. 

As regards waves, there are two extreme ways in which a 
perfect conductor behaves that is, extreme forms of the gene- 
ral behaviour. It may wholly conduct them, or it may wholly 
reflect them. In the latter case we may illustrate by ima- 
gining a thin plane electromagnetic sheet, consisting of crossed 
electric and magnetic forces in the ratio given by E = /*vH, 
moving at the speed of light, to strike a perfect conductor 
flush that is, all over at the same time, by reason of parallel- 
ism of the sheet and conducting surface. The incident sheet 
is at once turned into another plane sheet, which runs away 
from the conductor as fast as it came. If the conductivity be 


of the electric kind the reflected sheet differs from the incident 
in having its displacement reversed, but in no other respect* 
This is perfect reflection with reversal of E. During the act 
of reflection, whilst the incident and reflected sheets partly coin- 
cide, E is zero and H is doubled. Both are tangential ; but 
there can be no tangential E, so the reflector destroys E and 
initiates the reflected sheet, in which H is the same as in the 
incident sheet, whilst E is reversed. 

On the other hand, when the conductivity is of the magnetic 
kind, the reflected wave sheet differs from the incident only in 
having its induction reversed. The displacement persists, 
being doubled during the act of reflection, whilst the induction 
is then annulled. 

The other extreme occurs when a plane electromagnetic sheet 
hangs on to a conductor perpendicularly. It then slips along 
the conductor at the speed of light, with perfect slip. This 
may occur with a plane reflector, but, of course, the most 
striking and useful and practical case is that of a straight 
cylinder a wire, in fact, though it need not be round, but may 
have any form of section. The wave then runs along the wire 
at. constant speed v, and without change of type, at least so 
long as the wire continues straight and of unchanged section. 
If the section vary regularly, so that the wire is a cone, then it 
is a spherical wave that is propagated along it without change 
of type. This case includes an infinitely fine wire, when we 
may have either spherical or plane waves. Other interesting 
cases may be made up by varying the angle of the cone, or 
using a double cone, or a cone and a plane, &c. 

Now, in the first main case of perfect reflection (flush) the 
incident and reflected sheets are wholly separated from one 
another, except just at the reflecting surface, where there is 
a momentary coincidence. On the other hand, when a plane 
wave runs along a wire, or, say, more conveniently here, along 
a plane, we only see one wave. It is the case of reflection at 
grazing incidence, and may be considered the limiting case of 
permanent union of the incident and reflected waves. Between 
these two cases we have the general case of incidence and reflec- 
tion at any angle. There are two plane waves (sheets, most 
conveniently for reasoning and description) one going to and the- 
other coming from the plane reflector, where they join together,. 


making equal angles with it. In the overlapping region, close 
to the plane, the displacement due to the union of the two- 
waves is normal to the surface (that is, with an electric con- 
ductor) which is electrified, and the electrification runs along 
the surface at a speed depending upon the angle of incidence,, 
being v at grazing incidence (of rays) and v/cos 6 at incidence 
angle 0, varying, therefore, between v and infinity. It may be, 
perhaps, rather a novel idea to some readers that electrification 
can run through space at any speed greater than that of light, 
but the matter is made simple enough by considering the rate 
of incidence upon the reflecting surface of different parts of the 
plane sheet. In the case of nearly flash incidence of a sheet, 
its different parts strike the surface nearly simultaneously, so- 
that there is an immensely great speed of motion of the elec- 
trification along the surface. The electrification is the same in 
amount always, and is continuously existent, so we are some- 
what justified in speaking of the electrification moving ; but 
we may equally well regard it as a case of continuous genera- 
tion of electrification at one end and of annihilation at the 
other end of the part of the conducting surface which is 
momentarily charged, the generation and annihilation being 
performed by the different parts of the incident sheet and the 
reflected sheet as they reach and leave tLe surface. Details of 
these simple cases, leading to a plainer understanding, will 
come later, when these general notions are got over. In the 
limiting case of normal incidence of rays, when the incident 
sheet strikes the surface flush, the electrification is non-existent, 
It goes out of existence just as its speed becomes infinite. 

The above describes one extreme kind of reflection of a plane 
sheet at any angle, and is what occurs when H in the incident, 
and, therefore, also in the reflected wave, is tangential to the 
reflector, whilst E is in the plane of incidence. But when it 
is H in the incident wave that is in the plane of incidence,, 
and E is tangential, we have quite another kind of composition 
in the overlapping part near the reflector. There is no E within 
it at all, and also no electrification ; whilst the H within 
it is parallel to the reflector, and simply joins together the H's 
in the parts of the waves which do not overlap, the H in one 
wave being directed towards the surface in the plane of inci- 
dence, and in the other away from the surface. In both waves, of 


course, H is in the plane of the wave. In other respects we have 
a similarity in propagation. It is now a surface current, instead 
of electrification, that runs along the surface at speed v/cos 0. 

If the reflector be a magnetic conductor, we have two very 
similar main cases, in one of which magnetification (the ana- 
logue of electrification), and in the other a magnetic current, 
runs, or appears to run, along the surface. 

The Effect of Conducting Matter in Diverting External 

191. The theory of the effect of a finite degree ot con- 
ductivity in the medium on electromagnetic waves is far more 
difficult and complicated than that of the effect of infinite con- 
ductivity. Nevertheless, we may gain a general idea of the 
nature of the effect by means of the substitution of simple 
problems for the real ones that present themselves, and also in 
this way obtain a knowledge of very important properties con- 
cerned. There are two ways in which we may regard the ques- 
tion of conductivity. First, starting from the theory of perfect 
conductors surrounded by perfect dielectric non conductors, we 
may examine the effect of introducing a slight amount of 
resistivity, to be then increased more and more until at last 
we come to conductors of high resistivity, or infinite, when 
we have dielectrics merely. The other way is to start with 
electromagnetic waves in a perfect dielectric, and examine the 
effect produced upon them of introducing first a small amount 
of conductivity, then more and more, until we come to perfect 
conductors again. Both ways are instructive none the less 
because they give very different views of the same matter. 

In the first place, it may be readily conceived that if a con- 
ductor have only a slight amount of resistivity it may behave 
approximately the same as if it had none, and may obstruct 
waves nearly perfectly internally, and likewise reflect and con- 
duct them superficially nearly perfectly. This is true, but the 
element of time has to be taken into account, as it becomes of 
great importance when there is some resistivity, however little, 
instead of quite none. Suppose, for example, we have an 
initially steady magnetic field, and bring a conductor into it 
very quickly from a distance, and keep it there. If this be 
done quickly enough, the first effect is nearly the same as if 


the conductor were perfect. That is, the induction of the 
field will be driven out of the space it previously occupied, 
which is now occupied by the conductor, and pass round 
it tangentially, as if the conductor were of zero induc- 
tivity. It will, therefore, be mechanically acted upon by 
the stress in the field in a manner resembling the action 
upon a perfect diamagnetic body that is, there will be a 
force tending to drive it from stronger to weaker parts of the 
field. In another form, we may say that external force must 
be applied to the conductor to bring it into the field and 
to hold it there. But this state of things will not continue. 
The magnetic force, which is at first only skin deep, will pene- 
trate into the interior in time, according to a law resembling 
that of the diffusion of heat. Given time enough, it will 
assume the same distribution as if there were no conductivity, 
although our assumption is that there is nearly no resistivity 
that is, in the ultimate state tended to, it is merely the induc- 
tivity that settles how the magnetic induction will distribute 
itself. Initially, there is a skin current, its total being mea- 
sured by the difference in the magnetic force just outside and 
a little way inside the conductor, in the bulk of which there 
is practically no magnetic force or electric current. As in the 
-case of a perfect conductor, Ohm's law is fully obeyed. There 
is no internal current, because there is no internal electric or 
magnetic force. They simply have not had time to get in, on 
account of the obstructive action of the high conductivity. It 
is as useless and misleading to say that one electric force is 
-cancelled by another, as it is to ascribe the absence of magnetic 
force to the counteracting magnetic force of the skin current. 
As the magnetic force spreads into the conductor, so does the 
electric current, which is the curl of the former. In the end, 
when the magnetic force has got steady, the current ceases. 
There may now still be moving force on the conductor, but not 
of the same kind as before, it being simply the ordinary para- 
anagnetic attraction or diamagnetic repulsion according as the 
value of the inductivity of the body exceeds or is less than 
that of the external medium. 

It will be very much the same thing if we start with the 
conductor at rest in a neutral state in a neutral field, and 
then establish a steady magnetic field by some external 


cause. There will still be the same internal obstruction 
offered by the conductor, and consequent delay in the assump- 
tion of the steady state throughout it. In both cases, too, 
there is a necessary waste of energy involved in the process, 
according to Joule's law of the generation of heat by the exist- 
ence of electric current in conducting matter. Thus two things 
happen when the degree of conductivity is not infinite. First, 
the reflection is imperfect at the boundary of the conductor, 
a portion of an incident disturbance being transmitted into 
it. Next, in the act of transmission and the attenuation 
involved, there is a loss of energy from the electromagnetic 
field. We shall see later more precisely the nature of the 
attenuating process. At present we may note that if the 
external field be not steady or do not tend to a steady state, 
so that the conductor is exposed to fluctuating forces, then the 
internal part of the conductor need never acquire any sensible 
magnetic force. Thus, if the external field be the sum of a 
balanced alternating, and of a field which would be steady in 
the absence of the conductor, only the latter part will penetrate 
fully into it. The former alternating part will penetrate 
imperfectly, the more so the greater the conductivity, and, as 
before said, not at all when the conductivity is perfect. The 
other field is then also excluded. With rapid alternations the 
region of sensible penetration is only skin-deep, consisting of 
layers of opposite kinds (as regards direction of the magnetic 
force, which is nearly tangential), with, however, so very rapid 
an attenuation of intensity in going inward that practically 
only one wave need be considered (except for short waves, like 
light). Of course, the heat generation is now confined to the 
skin. What goes in further does so by ordinary heat diffusion. 

The time-constant of retardation of a conductor varies as the 
conductivity, as the inductivity, and as the square of the linear 
dimensions. This refers to the intervals of time required to 
establish a definite proportion of the steady state under the 
action of steady forces in bodies of different size, conductivity, 
and inductivity, but geometrically similar. Here the two pro- 
perties, conductivity and inductivity, act conjointly, so that, for 
example, iron is far more obstructive than copper, although its 
conductivity is much inferior. It is different with the heat- 
generation. There the inductivity and conductivity act in 


opposite senses, for, with the same electric force, the waste 
varies as the conductivity, or, with the same current-density, 
as the resistivity. It results that in cases of skin-conduction 
of rapidly alternating currents, the resistance per unit area of 
surface varies directly as the square root of the product of the 
resistivity (not conductivity), inductivity, and frequency. 
Thus, whilst we may increase the resistance by increasing the 
frequency, with a given material, and also by increasing the 
inductivity, we decrease it by increasing the conductivity, in 
spite of the fact that the internal obstruction varies as the con- 
ductivity and inductivity conjointly. The point to be attended 
to here is that mere internal obstruction is no necessary bar to 
effective skin conduction, although, of course, in a given case 
the resistance is greater than if the conduction were more wide- 
spread. It depends upon how it is brought about, whether by 
conductivity or inductivity. This is how it comes about that 
with the complete internal obstruction of a perfect conductor, 
with the effective skin reduced to nothing, there is still no 
resistance, and the slip of electromagnetic waves along them is 
perfect. But it is different when we obtain the internal obstruc- 
tion by increasing the inductivity, preserving the conductivity 
constant. Perfect internal obstruction then means infinite 
resistance, and no proper slipping of waves at all. If the 
obstruction be not complete, it will be accompanied by very 
rapid attenuation of waves running along the surface when the 
obstruction arises from high inductivity, and by relatively very 
slight attenuation when it arises from high conductivity. 

The repulsive force which was referred to in the case of a 
perfect conductor brought into a magnetic field, or when a 
magnetic field is created outside the perfect conductor, arises 
from its obstructive action, combined with the fact that it is 
only the lateral pressure of the magnetic stress that acts on the 
conductor, owing to the tangentiality of the magnetic force. 
This repulsive force is naturally also operative, though in a less 
marked form, when the conductivity is not perfect. In fact, it 
is operative to some extent whenever there is a sufficiently rapid 
alternation of the field for the conductance to cause a sensible 
departure from the undisturbed state of the magnetic force, 
and is, therefore, strongly operative with ordinary metallic 
conductors with quite moderate frequency of vibration. The 


remarkable experiments of Prof. Elihu Thomson on this " elec- 
tromagnetic repulsion " will be remembered. The phenomenon 
has nothing specially to do with electromagnetic waves. It is 
magnetic repulsion, rather than electromagnetic, using the 
word " magnetic " in its general sense, apart from the special 
fact of magnetisation when the iiid activity of the conductor 
is not the same as that of the ether.* 

When a conductor is brought into an electric instead f a 
magnetic field, the case is somewhat different. There is in 
any case merely skin conduction, for there is an actual destruc- 
tion of the flux displacement by conductivity. The final result 
therefore, is that the ultimate permanent state in the con- 
ductor is a state of perfect neutrality, just as if the conduc- 
tivity were perfect. Electric conductivity destroys displace- 
ment, but it cannot destroy induction. Similarly, magnetic 
conductivity would destroy induction, but would be unable to 
destroy displacement. Thus, a magnetic conductor brought 
into an electric field would, in time, permit its full penetra- 
tion, but if brought into a magnetic field the final result would 
be a state of internal neutrality, however low the conductivity. 
If, however, it be very low, then, whether it be of the electric 
or the magnetic kind, there will be an initial nearly complete 
penetration (owing to the removal of the obstruction), followed 
by subsidence to zero of the flux appropriate to the conduc- 
tivity, electric or magnetic respectively. The persistence of 
magnetic induction, in spite of the presence of electric con- 

* This reference to Elihu Thomson's experiments must not be under- 
stood as a full explanation, which is sometimes complex. The idea in the 
text has been of a lump of metal. When made a disc or a linear circuit 
we have special peculiarities, and the theory may be perhaps best done 
in terms of inductances and resistances. The principle concerned of 
the temporary diversion of magnetic induction by conducting matter 
remains in force, however, whether the matter be in a lump or in a closed 
line. In the latter case the tendency of the conductance is to keep the 
total induction through it constant. Consider first an infinitely con- 
ducting disc which completely diverts induction ; and next, a ring made 
by removing nearly all the disc except the outer part. Induction now 
goes through the circuit, of course, when brought into a magnetic field, 
but its total is zero, by reason of the infinite conductance and the current 
" induced " in the ring. As is well known, Maxwell considered perfectly 
conducting linear molecular circuits in applying his views to Weber 'A 
theory of diamagnetism. 


ductivity, is a very important and significant fact, of which I 
shall give a simple proof in a later Section, in amplification of my 
proof of 1887-8. The present method of passing from perfect 
to finite conductivity is unsuitable for the purpose, because 
in good conductors the dielectric permittivity is altogether 
swamped, and is therefore ignored ; whilst, on the other hand, 
in very bad conductors the permittivity is a factor of the 
greatest importance. Now we can pass continuously from a 
non-conducting dielectric to a conducting one, up to perfect 
conduction, but we cannot pass the other way without having 
the permittivity in view all the time, which makes the matter 

Parenthetical Remarks on Induction, Magnetisation, Indue- 
tivity and Susceptibility. 

192. As people's memories are very short, and there is some 
discussion on the subject, I may repeat here that the so- 
frequently-used word inductivity is not intended for use as a 
mere synonym for permeability. The latter is the ratio of the 
inductivity of a medium to that of ether, and is therefore a 
mere numeric. On the other hand, inductivity has a wider 
meaning, namely, such that j/*H 2 is the density of the magnetic 
energy, irrespective of dimensions. We can only make it a 
numeric by assumption. Even then, it has only a fictitious 
identity with permeability a forced numerical identity. 
Similar remarks apply to some other quantities, but they 
are particularly necessary in the case of inductivity, on 
account of the obscure and misleading manner in which the 
connections between induction, magnetic force, and magnetisa- 
tion were formerly commonly presented (and still are some- 
times), together with the misleading connection between the 
susceptibility and the permeability. We should write 

if p is the permeability and K the susceptibility, instead of 

where the suffix letter refers to the common irrational reckon- 
ing. The 4?r is, as usual, simply nonsense, unworthy of scientific 
men near the end of the nineteenth century. Now introduce 


Maxwell's ether theory, and make /? = /V/V where // is the 
inductivity of ether, and p that of some other substance (with 
the usual reservations), then 

takes the place of the common 

-which is misleading in two respects, first as regards the obstre- 
perous 47T, and next in making p and K be quantities of the 
same kind. But K is always and essentially a numeric, whilst //, 
is not. We see that the use of inductivity rather than permea- 
bility is necessary in electromagnetic theory, as a matter of 
logical common sense as well as for the purpose of scientific 
clearness. But this need not interfere with the use of permea- 
bility in its above-described sense of a ratio. If, on the other 
hand, one of the two words should be abolished, there can, I 
think, be little doubt as to which should go. 

In the case of purely elastic magnetisation (without intrinsic) 
we have 

where B is the induction and F the magnetic force. Here 
is what the induction would be in ether, so that the additional 
part /* KF expresses the effect of the matter present, which 
becomes magnetised. The ratio K-, therefore, naturally expresses 
the susceptibility for magnetisation of the matter. Perhaps, in 
passing, it might be thought that /A O K should express the sus- 
ceptibility. But this will not do, because magnetisation and 
induction are similar. The induced magnetisation is />t KF. In 
strictness it should not be called the intensity of magnetisation, 
but rather the density, if we properly carry out Maxwell's prin- 
ciples about forces and fluxes, or intensities and densities. B is 
a flux, therefore so is /* KF, to be measured per unit area. 
Now, the common form is 

or i = i + 7r i , 

if 1. = K^i. Here we have, apart from the 4?r absurdity, an 
irrationality of a different kind, viz., that induction and mag- 
netisation are made identical in kind with magnetic force, since 
we have the difference of two flux densities expressed by an 


"intensity," which is referred to unit length. Now, this may 
matter very little in practical calculations, but it is more than 
mischievous in theory. Suppose, for example, we are working 
with a kind of mathematics that takes explicitly into account 
the two ways of measuring vector magnitudes, with reference 
to length and to area respectively, according as they are re- 
garded as " flux " densities or " force " intensities. Then, if we 
do not recognise and take account of the radical difference 
between B; and F f in the last equation, we may expect to be led 
to singular and unaccountable anomalies. This is, I think, 
what has happened in Mr. MacAulay's recent paper on the 
theory of electromagnetism. The remedy is easy. There 
should be no special limitations imposed upon the quantities 
concerned such as occur when permeability and inductivity are 
made the same. 

It is also highly desirable, for the same purpose of obtaining 
scientific clearness and freedom from distressing anomalies, 
that the distinction between " induced " and intrinsic magneti- 
sation should be clearly recognised and admitted in the formula. 
In the above there has been no intrinsic magnetisation. Let 
this now be I . An equivalent form is /xh , where h is the cor- 
responding intrinsic magnetic force. This I is of the same 
nature as B. The complete induction becomes 

where H is the complete force of the flux B. This is the best 
way of exhibiting the induction. If H be split at all, let it be 
into the part involving the intrinsic force h and the rest. Or, 

The other separation, namely, of //F into the ether induction 
/x F and the additional part due to matter, is less useful. If 
done, then 

If we now amalgamate I and I, to make, say, I 1} the total 
magnetisation, intrinsic and induced, we have 

which, translated into irrational units, makes 


and lastly, omitting the /t* by assuming it to be unity, we 
obtain the common 

B i -F < +47rI li , 

containing three faults, the arbitrary 47r, the equalising of an 
intensity and a flux, and the unnatural union of physically dis- 
tinct magnetisations. 

" Different men have different opinions some like apples, 
some like inions." But can anyone possibly really like the 
roundabout and misleading way of presenting the magnetic 
flux relations which I have above criticised? There is no 
excuse for it, except that it was employed by great men when 
they were engaged in making magnetic theory, before they 
had assimilated its consequences thoroughly. When the rough 
work of construction is over, then it is desirable to go over 
it again, and put it in a better and more practical form. We 
should copy the virtues of great men, if we can, but not their 

Men who are engaged in practical work can hardly be expected 
to fully appreciate the importance of these things, because their 
applications are of so highly specialised a nature, in the details 
of which they may become wholly absorbed. They may even 
go so far as to say that the paper theory of magnetic induction is 
not of the least moment, because they are concerned with iron, 
and although there may be a certain small range of application 
of the theory even in iron, yet they are scarcely concerned 
with it, and, therefore, it is of no consequence. There could 
not be a greater mistake. A complete theory of magnetic 
Induction, including hysteresis, must necessarily be so con- 
structed as to harmonise with the limited theory that has 
already been elaborated, which is understandable when exhibited 
in a purified form, freed from 47r's and other anomalies. First 
of all, we have the ether, in which B = /* F or B = /x H, because 
of the absence of intrinsic magnetisation. Next we come to 
elastically magnetised bodies in which the relation between flux 
and force is linear. Then B = /xH, where ^ differs from /x , being 
either greater or smaller, and is either a constant scalar, or else 
(with eolotropy) a linear operator. If there is no intrinsic mag- 
netisation, F and H are still the same, and the curl of either is 
the current density. But should there be intrinsic magnetisa- 
tion, then H = h ( , + F, whilst B = /xH still, and it is the curl 


of P that is the current density. Or, B = I + /*F. The next 
.step is to make fiF be not a linear, but some other function 
of F, to be experimentally determined, if it be possible to 
express B - 1 , which is the free induction, as a function of F. 
Of course, it can be done approximately. Then comes the 
difficult question of hysteresis. This involves the variation of 
I with F, with consequent waste of energy. If this little 
matter be satisfactorily determined, we may expect to have a 
sound mathematical theory of magnetic induction in an 
extended form which shall properly harmonise with the 
rational form of the elementary theory. The divergence of 
B is zero all through. The experimental justification of this 
generalisation is the fact that no unipolar magnets have yet 
been discovered. 

Effect of a Thin Plane Conducting Sheet on a Wave. 
Persistence of Induction and Loss of Displacement. 

193. Coming now to the effects produced on electromag- 
netic waves by a small amount of conductivity, to be after- 
wards increased, we shall adopt a particular device for simplify- 
ing the treatment. Imagine, first, the dielectric medium to 
possess a uniformly-distributed small conductivity. Evidently, 
che action of the conductivity on a wave is a continuous and 
cumulative one. Next, localise the conductance in parallel 
sheets that is, substitute for the uniform conductivity a great 
number of parallel plane conducting sheets, between which the 
medium is non-conducting. If we increase their number suffi- 
ciently, their action on a wave whose plane is parallel to that 
of the sheets will approximate, in the gross, to the effect of the 
uniform conductivity which the conductance of the sheets 
replaces. We have, therefore, to examine the influence of a 
single very thin conducting sheet upon a wave. This is not 

Imagine, then, a simple plane electromagnetic sheet to be 
running through the ether at the speed of light. This is the 
natural state of things ; and, in the absence of conductivity or 
other disturbing causes, there will be no change. Now insert 
a plane conducting sheet in the path of the wave. It should 
be so thin that the retarding effect of diffusion within it is 
quite insensible. Let the wave strike it flush. The theory 



shows that it is immediately split into two similar waves, one 
of which is transmitted beyond the plate, whilst the other is 
reflected. The transmitted wave differs from the incident in 
no respect except strength. It is attenuated in a certain ratio 
depending upon the conductance of the sheet, being greatlv 
attenuated when the conductance is large, and slightly atten- 
uated when it is small. The formulee are reserved. This 
transmitted wave moves on just as the incident wave did, and 
nothing further happens to it. The reflected wave, on the 
other hand, having its direction of motion opposite to that of 
the incident and transmitted waves, travels back the way it 
came, and nothing further happens to it. 

The direction of the magnetic force in the three waves is the 
same. This is one general property, irrespective of the amount 
of conductance. But a much more striking one connects the 
intensity of the magnetic force in the three waves. The sum 
of the intensities in the reflected and transmitted waves equals 
the intensity in the original incident wave. That is, the con- 
ductance, with dissipation of energy, has had no effect whatever 
on the total induction. It has merely redistributed it, by 
splitting it into two parts, which then separate from one 
another. The " number of tubes " in the reflected wave may 
be made to bear any ratio we please to the number in the 
transmitted wave by altering the conductance of the plate : 
but their sum is always exactly the number of tubes in the 
incident wave. This property exemplifies, in a manner which 
may be readily understood, the persistence of induction, in 
spite of conduction and waste of energy. 

But as regards the displacement, the case is quite different. 
From the fact that the reflected wave runs back, whilst its 
magnetic force preserves its original direction, we see that the 
electric force must be reversed. On the other hand, it is 
unchanged in the transmitted wave. If, then, their sum 
equalled the electric force in the incident wave, it would imply 
that the transmitted wave was of greater amplitude than the 
incident. But it is smaller, invariably. So there is a loss of 
displacement. Thus, if H in the incident becomes (1 - 7i)H in 
the transmitted wave, where n is some proper fraction, it 
becomes nH in the reflected wave. At the same time E in the 
incident becomes (1 -?t)E in the transmitted, and - wE in the 


reflected wave. The loss of E is, therefore, 2nE, or the loss of 
displacement is 2nD. By the loss we mean the excess of the 
displacement in the incident over the sum of the displacements 
in the two resulting waves, transmitted and reflected. This 
loss occurs at the very moment the incident wave coincides 
with the plate. The plate itself may be regarded as a 
dielectric homogeneous with the ether outside, or perhaps of 
different permittivity, but with the conducting and dissipating 
property superposed. When thin enough, the permittance of 
the plate is of insensible influence, and may be disregarded. 
But strictly, a conducting dielectric is a dielectric which cannot 
support displacement without wasting it, so that a continuous 
supply of fresh displacement is needed to keep it up. The rate 
of waste of energy is proportional to the electric stress. 

But it should be carefully noted that the loss of energy and 
the loss of displacement are entirely distinct things, which are 
not proportional ; and that the loss of displacement itself may 
sometimes require to be understood in a somewhat artificial 
sense. For the loss may be greater than what there is to lose. 
It must then be understood vectorially. This occurs when n 
is greater than \. When n = J, the reflected and transmitted 
waves are equally strong, and only differ in the direction 
of the displacement. The loss of displacement is, there- 
fore, complete. The loss of energy in the plate is simul- 
taneously at its maximum, being equal to one-half of the 
energy of the original wave. If we reduce the conductance of 
the plate, we increase the transmitted wave, and reduce the 
waste of energy in it and the loss of displacement. The 
amount of the latter still remains positive, therefore, assuming 
it to be positive in the incident wave. The extreme is reached 
when the plate has no conductance. Then the incident wave 
goes right through without any splitting and reflection, and 
therefore without attenuation, and there is no waste of energy. 
On the other hand, if we increase the conductance above the 
critical value making n = J, we reduce the transmitted wave 
and increase the reflected, whilst we simultaneously reduce the 
waste of energy in the plate and increase the loss of displace- 
ment. The extreme is reached when the plate is a perfect 
conductor. There is then no transmitted wave and no loss of 
energy, whilst the reflected wave is of full size, but with the 

AA 2 


displacement reversed as compared with the incident, so that 
the loss of displacement, in the sense described, is the greatest 
possible, viz., 2D. 

It will be seen from these details that whilst the absorption 
or dissolving of induction by conductance is a myth, the idea 
of an absorption of displacement is not without its incon- 
veniences when the conductance is great, and that this becomes 
extreme when there is really no loss of energy in the plate, 
when, in fact, the incident wave does nothing in it, but is wholly 
rejected with its displacement reversed. It seems, then, more 
natural to consider the waste of energy from the field caused 
by the plate as loss. This takes place equally from the electric 
and magnetic energies, since they are necessarily equal in every 
one of the three waves. But in the application made later, 
the plate is to have very slight conductance (in the limit an 
infinitely small amount), so that the total displacement cannot 
change sign, but merely suffers a slight loss. Then the idea 
of loss of displacement by conductance becomes useful again. 

The loss of energy takes place as the incident wave is travers- 
ing the plate. Its successive layers each cause a minute attenua- 
tion of the wave passing, and this applies equally to the induc- 
tion and displacement, so that the transmitted wave emerges 
from the plate a pure electromagnetic wave, a reduced copy of 
the incident. The successive layers, too, each cast back a 
minute portion of the wave traversing them, with unchanged 
sign of the induction, but with displacement reversed; and 
these rejected fluxes make up the reflected wave. There are 
evidently residual effects due to the internal reflections of 
minute portions of the main reflected wave, but these residuals 
tend to vanish when the plate is thin enough. 

If the plate be a magnetic instead of an electric conductor, 
the theory is quite similar. The transmitted wave is an 
attenuated continuation of the incident. The reflected wave 
is also a copy of the incident, also reduced. But it is now 
the induction that suffers loss, because its direction in the 
reflected wave is opposite to that in the incident and trans- 
mitted. On the other band, the displacement now fully per- 
sists, being merely split into two parts by the plate. 

Notice that if the plate be both an electric and a magnetic 
conductor, its attenuating effect from these two causes on the 


transmitted wave will be additive, so that it will emerge a 
pure wave with extra attenuation. But as regards the reflected 
wave, we have a peculiar result. The action of the magnetic 
conductance is to reverse the induction whilst keeping the 
displacement straight ; whilst that of the electric conductance 
is to reverse the displacement and keep the induction straight. 
The result is that the reflected wave is reduced in magnitude 
by the addition of magnetic conductance to previously existent 
electric conductance. With a proper proportioning of the two 
conductances, the reflected wave may be brought nearly to 
evanescence from a plate of finite conductance. In the limit 
the compensation is perfect, and the incident wave goes right 
through without reflection, though it suffers extra attenua- 
tion. This is the explanation of the distortionless propagation 
of waves in a dielectric medium possessing duplex conducti- 
vity, electric and magnetic. Whilst there is no reflection in 
transit, there is a continuous loss both of displacement and of 

The Persistence of Induction in Plane Strata, and in general. 
Also in Cores and in Linear Circuits. 

194. Now return to the case of electric conductivity alone, 
and, as described, let it be locally condensed into the conduc- 
tance of any number of parallel plates. We know that the 
effect of any one of them on a thin electromagnetic sheet is to 
split it, as previously described. If we like, therefore, we can 
follow each of the resulting waves, and observe how they are, 
in their turn, split by the first plates they meet, giving rise to 
four waves, to be a little later split into eight, and so on. This 
process may seem cumbrous, but it is also an instructive one. 

Thus, consider what happens to the total induction. We 
know that it persists in amount and direction when a single 
split occurs. Now the same property applies to every succes- 
sive split a wave suffers in our dielectric medium containing 
parallel conducting plates. So the total induction remains 
constant. It is redistributed and spreads out both ways, but 
without the least loss. There is a small loss of energy at every 
split, but this does not affect the total induction. This applies 
when we start from a single pure electromagnetic sheet moving 
either way. It therefore applies when the initial state consists 


of any number of such sheets, of any strengths, forming a per- 
fectly arbitrary initial distribution of induction and displace- 
ment in parallel plane layers. There is still persistence of 
the induction. Finally, the same applies when we split up 
the conducting plates themselves into plates of smaller con- 
ductance, and spread them out at uniform distances. The 
ultimate limit of this process is reached when the conduct- 
ance is quite uniformly spread, so that we have a perfectly 
homogeneous medium under consideration. It is, fundament- 
ally, a dielectric propagating disturbances at speed v ; but it 
is, in addition, a conductor as well, and distorts the waves and 
dissipates their energy. The speed v is (MC)"*, with the proper 
values of //. and c. The conductivity does not interfere with 
this property of propagation at finite speed. But observe that 
if we choose to ignore the displacement, then the corresponding 
speed is infinitely great. We conclude from the above that 
plane sheets of induction in electric conductors always preserve 
the total induction constant in amount, irrespective of the 
amount of elastic displacement, or whether there is any at all. 
That is, induction cannot be destroyed by conductance. 

If, then, it suffers destruction, this must be due to some 
other cause. It may be merely a cancellation by the union of 
oppositely-directed inductions. This may be termed a vectorial 
cancellation. It may occur, of course, with plane strata of 
induction. Thus if, in an infinitely large conductor, the total 
induction be initially zero, which does not require the induction 
density to be zero, the final effect will be a complete annihila- 
tion of the induction by mutual cancellations. Should, how- 
ever, the total induction be not zero, it will persist. The 
induction density will tend to zero, but that will be merely on 
account of its attenuation by spreading, not because there is 
any destruction by the conductance or resistance when either 
of them is finite. To prevent the attenuation to zero we may 
interpose infinitely conducting barriers, one on each side, in 
planes parallel to the sheets of induction. Then the final 
result will be that the induction will spread itself out uniformly 
between the barriers and maintain a finite density, 

To illustrate this property in a somewhat less abstract man- 
ner, consider a large ring, say of copper, though iron will do 
equally well except as regards some complications connected 


with its magnetisation. Let it be inductised by an enveloping 
coil-current so that the induction goes along the core in a 
complete circuit. When it is steadily set up, if we remove 
the coil- current (and the coil too, preferably for our present 
purpose) the induction in the core will, in time, all come out of 
it. But if we clap an infinitely-conducting skin upon the core, 
it will not come out. Then we have a certain flux of induction 
locked up, as it were, in a conducting material, which has no 
effect upon it. It can neither be destroyed by the conductance 
of the core nor can it get through the perfectly-obstructive 
skin. If the skin is clapped on after the induction has 
partially escaped (which escape begins on the outside, before 
the interior is sensibly affected), there is a redistribution of 
induction, which continues until a new state of equilibrium is 
reached. During this process there is electric current in the 
core and some waste of energy. But there is no waste of the 
induction. The final induction is the mean value of .the 
original induction across the section of the core. 

In further illustration, let the core be hollow and be induc- 
tised circularly instead of along its length by means of two 
currents on its boundaries, inner and outer, oppositely directed, 
following the length of the core. When this is done, remove 
the currents and clap on perfectly conducting skins internally 
and externally. There will be a similar persistence of the 
induction, although its tubes now go round the inner boundary 
circularly. There may be an initial settling down, but the 
outer skin will not let the induction expand outwardly, and the 
inner skin will not let it contract inwardly. If the latter could 
happen we might have cancellation. To get this effect remove 
the inner skin. Then, whether we fill up the hollow with 
finitely conducting matter or leave it nonconducting, we allow 
the induction to spread internally and permit cancellation. The 
induction will now wholly disappear, in spite of the external skin. 
That is to say, there will be a continuous passage of the induc- 
tion out of the initially inductised region, accompanied by elec- 
tric current therein, which will continue until the whole of the 
magnetic energy is wasted as heat in the core. 

The same property is exemplified, though in a less easily under- 
standable manner, with a single closed line or circuit of infinite 
conductance. If it embrace a certain amount of Induction it 


will always do so, in the absence of impressed force to alter 
the amount. The induction is locked in and cannot pass 
through the infinitely- conducting circuit to dissipate itself. 
If the conductance of the circuit be finite, then it can get 
through. The time-constant varies as the conductance. The 
disappearance of the induction is manifested by the waste of 
energy in the circuit, the electric current in which is sup- 
ported by the voltage of the decreasing induction through it. 
But the current is there all the same (measured magnetically) 
when the conductance is infinite. The induction is steady, 
and there is no voltage in the circuit. But none is needed. 

On the other hand, if there is initially no induction through 
the circuit, there will continue to be none when a magnetic field 
is created in its neighbourhood. But although the tubes of induc- 
tion cannot cut through the infinitely conducting circuit so as to 
make the induction through it be a finite quantity, yet they do 
pass through a surface bounded by the circuit, as much positively 
as negatively. The resulting induction distribution is to be got 
by superimposing the external induction and that due to a cur- 
rent in the circuit of such strength as to make the total induction 
through it be zero. The property is a general one, for if the cir- 
cuit be moved about in a magnetic field, there is always, in virtue 
of its impermeability to the magnetic flux, zero total induction 
through it if its conductance be infinite ; whilst if it be finite 
but great, there is an approximation to this result so long as the 
motion is kept up, or the external field be kept varying. At the 
same time, the least amount of resistance in the circuit will be 
sufficient, if time enough be given, to allow the external induc- 
tion, when due to a steady cause, to get past it to the full 
extent, when of course the current in the circuit will cease. 

In a similar manner, displacement can be locked up by a 
circuit of perfect magnetic conductance. There is also per- 
sistence of displacement in spite of a finite degree of magnetic 
conductivity in a continuous medium, unless it" be electrically 
conducting as well. 

The Laws of Attenuation of Total Displacement and Total 
Induction by Electric and Magnetic Conductance. 

195. Next consider the effect of a conducting medium upon 
the total displacement. We know that the latter decreases 


with the time, and the law of decrease may be readily found 
from the theory of a single conducting plate. We found that 
when its conductance exceeded a certain value, the loss of dis- 
placement exceeded the original. But, in regarding the action 
of a homogeneous conductor upon a wave as the limit (in the 
gross) of that of an assemblage of parallel plates in which the 
conductance is localised (which process may not seem unassail- 
ably accurate beforehand, but which is justified by the results), 
it is easy to see that we have merely to deal with plates of 
such very low conductance that the loss at each is extremely 
small, so that the above-mentioned difficulty does not enter. 
Thus, let the loss at one plate be such as to reduce the initial 
displacement D in a wave to mD, where m is a fraction nearly 
equal to unity. Here mD is the sum of the displacements in 
the transmitted and reflected waves, the latter being very 
small and of the opposite sign to the initial D. As these waves 
separate, they reach other plates and are split anew. If these 
plates have each the same conductance as the first, the total 
mD is further attenuated by them to w 2 D when the two waves 
become four. Next, when these four waves are split into eight 
by the next plates that are reached, the total displacement 
becomes m 3 D ; and so on. These successive displacement totals 
decrease according to the law of a geometrical series. It fol- 
lows that, in the limit, we shall have the total displacement 
represented by an exponential function of the time, say by 

D = D oe -<, (1) 

where D is the initial value, and D what it becomes at time t. 
To find the value of n, we have merely to examine the form of 
the fraction m, observe how it depends on the conductance of 
one plate, and proceed to the limit by making the number 
of plates infinite, whilst their conductances are infinitely small. 
The result is that the constant n has the value &/c, where k is 
the conductivity and c the permittivity of the homogeneous 
conducting medium. 

In the irrational units of the B.A. Committee this quantity 
is represented by 47r&/c, which is, of course, nonsense, like the 
quaternionic doctrine about the square of a vector. They are 
both going to go. The above reasoning applies to any initial 
distribution of displacement in plane layers, instead of merely 


one elementary sheet. Therefore, equation (1) shows that the 
total displacement subsides according to the time-factor 
e ~ ' c ' Now, this represents Maxwell's law of subsidence of 
displacement in a conducting condenser (apart from " absorp- 
tion" and hysteresis), or of the static distribution of displace- 
ment associated with electrification in a conducting medium. 
We see that the law has a far more general meaning. The 
initial displacement need not be static, but may be accom- 
panied by magnetic induction, and may consequently move 
about in the most varied manner, whilst its total amount 
decreases according to the static law. A homogeneous medium 
is presupposed, and modifications may be introduced by the 
action of boundaries. 

Passing next to the analogous case of a magnetic conductor, 
in which the total displacement remains constant whilst 
the total induction subsides, it is unnecessary to repeat the 
argument, but is sufficient to point out the law according to 
which the subsidence occurs. If B be the initial total 
induction, and B what it becomes at time t, we shall have 


where g is the magnetic conductivity and p. the indue tivity. 
The time-constant cjk of the former case has become p/g. 

Returning to the former case, it should be noted that when 
the initial distribution is of the static nature, unaccompanied 
by magnetic force, it retains this property during the sub- 
sidence. For, since the displacement subsides everywhere 
according to the same time-factor, its distribution does not 
alter relatively, or it remains similar to itself. Since, then, 
there is no magnetic force, there is also no true electric current. 
There is also no flux of energy. That is, the electric energy 
is converted into heat on the spot. 

A considerable extension may be given to this property. If 
there be a conducting dielectric in which the permittivity 
varies from place to place, containing a static distribution of 
displacement, then, if the conductivity vary similarly from 
place to place, so that the time-constant cjk is the same every- 
where, the displacement will subside everywhere alike, without 
magnetic force or flux of energy, and with purely local dis- 


sipatipn of the electric energy. For the solution is repre- 
sented by 

E = E , -**, H = 0, 

where E is the initial electric force of the static kind, having 
no curl, and E that at time t. Both the fundamental circuital 
laws are satisfied, the first because the true current is the 
sum of the conduction and displacement currents, and the 
second because E has no curl and k/c is constant. If it were 
not constant then, obviously, the property considered would not 
be true ; there would be different rates of subsidence at 
different places, and the distribution of displacement would 
change, along with magnetic force, electric current and 
transfer of energy. 

The corresponding property in a magnetic conductor 
requires the constancy of the time-constant fi/g. Then, 
whether p and g are themselves constant or variable from 
place to place, a static distribution of induction subsides 
everywhere alike, and without the generation of electric force. 

Returning again to plane strata of displacement in an 
electrically-conducting homogeneous dielectric, it may be 
inquired how the property (1) of the subsidence of the total 
displacement will be affected by the simultaneous existence of 
magnetic conductivity. This will undoubtedly affect the 
phenomena in detail, but will have no effect on the property in 
question. Similarly, the law (2) of the subsidence of total 
induction will not be affected by the presence of electric con- 
ductivity. That is, in general, when there are both conduc- 
tivities present, and both the fluxes displacement and induction 
present, the total displacement subsides according to one law 
and the total induction according to the other, without inter- 
ference. These properties have their parallels in the theory of 
telegraph circuits, as we shall see later. 

It should be remembered that we are dealing always with 
matter in the gross, and not with molecules at all ; or, equiva- 
lently, we assume a homogeneous constitution of the elements 
of volume. Thus, when displacement subsides in an electric 
conductor without generating magnetic force, the possibility 
and necessity of which are clearly indicated by the two cir- 
cuital laws, it may be that if we go in between the molecules 


there is magnetic force. It is, in fact, difficult to conceive how 
displacement in a heterogeneous medium of molecular consti- 
tution could be done away with without the generation of 
magnetic force, considering that the energy of the displace- 
ment is converted into heat energy. 

This matter, however, does not belong to the skeleton theory 
of electromagnetism, but is rather to be considered as a side- 
matter involving physical hypotheses to account for the influ- 
ence of matter upon the electromagnetic laws. 

The Laws of Attenuation at the Front of a Wave, due to 
Electric and Magnetic Conductance. 

196. Besides the above simple laws relating to the subsi 
dence of the total fluxes (sometimes true for the elementary 
parts) there are equally simple laws relating to the subsidence 
of the fluxes at the front of a wave advancing into previously 
undisturbed parts of the medium, which sometimes admit of 
extension to the body of the wave. To understand this it may 
be mentioned first, that the front of a wave in a non-conduct- 
ing dielectric is always pure ; that is, the electric and magnetic 
fluxes are in the wave-plane, and are perpendicular and in 
constant ratio. The body of the wave need not be of this 
pure type, owing to the change of form of the wave-front and 
other causes, but the property of purity always characterises 
the wave-front. This may be disguised in the case of a thin 
electromagnetic shell, when it is regarded as the front, for 
the shell itself may be complex. Then the mere front of the 
shell may be the only quite pure part. But taking cases 
free from this complication, we should next note that the 
introduction of conductivity into the medium makes no 
difference in the form of the wave-front or its position at a 
given stage of its progress, provided, of course, that the two 
quantities upon which the speed of propagation depends the 
inductivity and permittivity are not altered. Now, as has 
been already explained in connection with the theory of a thin 
conducting plate, as the wave advances through a continuously 
conducting medium its successive layers are being continuously 
subjected to a reflecting process, a minute portion of every 
layer being thrown back, whilst the bulk is transmitted. In 


the body of a wave, therefore, there is a mixed-up state of 
things. At the very front, on the other hand, there is no such 
mixture, for the disturbance consists wholly of what has been 
transmitted of the front layer. We may, therefore, fully expect 
that the law of its attenuation in transit is of a simple nature. 
To find it, locally condense the conductance into that of 
any number of equal conducting plates. Let any one of these 
plates attenuate a wave traversing it from E to wE. If 
initially pure it emerges a pure wave, and passes on to the 
next plate, where it suffers a second attenuation viz., to 
w 2 E, and again emerges pure. At the third plate it becomes 
w 3 E, and so on. The reflected portions we wholly ignore at 
present. The limit of this process, when the plates are in- 
finitely closely packed and of infinitely small conductance, so 
as to become a homogeneous dielectric possessing finite con- 
ductivity, is that the time-factor of attenuation takes the 
exponential form. The result is 

E = E e-*</ 2 <, (3) 

E being the initial, and E the value at time t. The time- 
constant 2c/k is just double that of the subsidence of total 
displacement. Whilst, for example, the total displacement in 
a plane wave attenuates to, say, y^- of its initial value, the 
disturbance at the wave front has only attenuated to j 1 ^ of its 
original value. 

The property (3) applies to the magnetic as well as to the 
electric force and flux. It does not apply merely to plane 
waves, but to any waves, because the superficial layer only is 
involved, and any elementary portion thereof may be regarded 
as plane. So it comes about that the exponential factor given 
in (3) makes its appearance in all investigations of waves in 
electrical conductors when the permittivity is not ignored. It 
is a more fundamental formula than the previous one with the 
time-constant c/&, which is the final result of the complex pro- 
cess of mixture of reflected waves, or is equivalent thereto. 

The corresponding property in a magnetic conductor is that 
the disturbance at the front of a wave is attenuated in time t 
according to the time-constant 2p/g. Thus, 



Here the time-constant is twice that of the subsidence of the 
total induction. Like the former formulae, these modified 
ones, (3) and (4), have their representatives in the theory of a 
telegraph circuit, in spite of the absence of magnetic con- 
ductance. It is replaced by something that produces approxi- 
mately the same result. 

In a conductor possessing duplex conductivity, electric and 
magnetic, their attenuative actions at the wave-front are 
independently cumulative, or additive. The attenuation is 
expressed by 

: ... (5) 

It is really the attenuative actions of a single conducting plate 
that are additive. This applies separately to every successive 
thin conducting layer through which the front of the wave 
runs, with the result (5), where the time-factor is the product 
of the two former time-factors of (3) and (4). 

In the theory of coils and condensers, not only do we meet 
with the time-constants L/R and S/K, the ratios of inductance 
to resistance and of permittance to conductance, but also with 
the double values. Their ultimate origin may be traced in the 
theory of the effect of a thin conducting plate upon a wave. 

The exponential time-factors concerned in (3) and (4), and 
the more complex one in (5), also make their appearance in 
connection with the disturbance in the body of a wave, though 
in a less simple manner. This will be returned to. 

The Simple Propagation of Waves in a Distortionless 
Conducting Medium. 

197. Coming now to the influence of conductivity on a 
wave elsewhere than at its extreme front, where we have 
recognised that the influence is simply attenuative, the easiest 
way of treating the matter is not to pass from the known to 
the unknown, but to reverse the process and pick out the 
cases which theory indicates are most readily understandable. 
This is to be done by a process of generalisation. The theory 
of a conductor with duplex conductivity is, in a certain case, 
far simpler than that of a real electric conductor. We have 
already mentioned that the reflective actions of two plates, 
one an electric, the other a magnetic conductor, are of oppo- 


site natures. The first reverses the displacement, and the 
second reverses the induction when throwing back a portion 
of the wave. The joint action of the two plates when 
coexistent and coincident, or the action of a single plate with 
duplex conductance, results in a complete disappearance of the 
reflected wave when the conductances are in proper ratio and 
the plate is infinitely thin. We then have transmission with 
attenuation but without reflection. This occurs, in a homo- 
geneous medium, when k/c=g/p. Reflex action being abolished, 
we are reduced to a kind of propagation of unique simplicity. 

To see the full meaning of this, start from any initial distribu- 
tions of induction and displacement in a non-conducting dielec- 
tric. Imagine that we have obtained the full solution showing 
the subsequent history of the disturbances. Now, if we introduce 
only one kind of conductivity, say electric, we shall, with the 
same initial state, have a profoundly different subsequent his- 
tory. Again, with magnetic conductivity alone, we shall have 
a course of events different from both the previous. But if we 
add on magnetic conductivity to previously existent electric 
conductivity, we shall partly counteract the distorting influ- 
ence of the latter. This counteraction becomes complete when 
the value of the magnetic conductivity is raised so high as to 
produce equality of the time-constants of attenuation due to 
the two conductivities separately. Further increase of the 
magnetic conductivity will overdo the correction and bring on 
distortion again, though of a different kind. 

Similarly, the distortion due to magnetic conductivity alone 
is diminished by introducing electric conductivity, and becomes 
completely abolished when there is enough of the latter to 
equalise the time-constants. Further increase brings on the 
distortion again, which is now of the electric kind. 

When the state of balance occurs, and the distortion is 
wholly removed, the course of events following any initial state 
is precisely the same as in a non-conducting medium, but with 
a continuous attenuation expressed by equation (5) above spe- 
cialised to suit the equality of the time-constants. That is, the 
time-factor of attenuation is now ~*^ c . This removal of dis- 
tortion applies to every kind of wave. 

This distortionless state in conducting media furnishes a 
sort of central basis for investigating the more recondite effects 


accompanying distortion. Nevertheless, its consideration would 
possess only a theoretical value, on account of the non-existence 
of the second kind of conductivity involved, were it not for the 
remarkable practical imitation of the distortionless state of 
things which is presented in the theory of telephone and other 
circuits under certain circumstances. If we abolish the ficti- 
tious magnetic conductivity throughout the medium traversed 
by the waves, we should, to have distortionless transmission, 
also abolish the electric conductivity. This is only to be 
attained by using wires of no resistance to guide the waves 
through a non-conducting medium. But they have resistance, 
of greater or lesser importance according to circumstances. Of 
what nature, then, is the distortion of waves produced by the 
resistance of a wire along which they run? The answer is, 
that it is approximately of the kind due to magnetic con- 
ductivity in the medium generally. On the other hand, 
the different kind of distortion due to electric ;onduc- 
tivity in the medium generally remains in action, being 
the effect of the leakage-conductance of the insulating 
medium surrounding the wire, or the average effect of other 
kinds of leakage at distinct and separate spots along the 
circuit. Thus we obtain an approximate reproduction of the 
theory of magnetic conductivity acting to neutralise the 
distorting effect of electric conductivity. The time-constants 
fj./g and c/k become L/R and S/K in a telegraph circuit, 
L being inductance, R resistance, S permittance, and K 
leakage-conductance. Their equalisation produces the distor- 
tionless circuit, which may turn up again later on. In the 
meantime I may remark that if the reader wishes to under- 
stand these things, he must give up any ancient prejudices he 
may be enamoured of about a " KR law " and the consequent 
impossibility of telephoning when "KR" is over 10,000. 
When pointing out, in 1887, the true nature of the telephonic 
problem and the absurdity of the " KR law " applied thereto 
generally, I predicted the possibility of telephoning with 
" KR " several times as great. It has since been done. In 
America, of course. A short time since, in noticing the KR = 
32,000 reached by the New York-Chicago circuit, I further pre- 
dicted that it would go up a lot more. It did very shortly after. 
The record is now about 50,000 (Boston-Chicago) for practical 


work, I believe. It means a good deal more for possible work. 
But there is no need to stop at 50,000. That can be largely 
exceeded in an enterprising country. 

The Transformation by Conductance of an Elastic Wave to 
a Wave of Diffusion. Generation of Tails. Distinct 
Effects of Electric and Magnetic Conductance. 
198. We are now prepared to somewhat understand the 
nature of the changes suffered by electromagnetic waves in 
transit through a conducting medium. It being the distortion 
due to the conductance alone that is in question, we eliminate 
that due to other causes by choosing plane waves for examina- 
tion, since these do not suffer any distortion in a homogeneous 
dielectric when it is non-conducting. Imagine, then, a simple 
electromagnetic plane wave-sheet of small depth to be running 
through a dielectric at the natural speed conditioned by its 
inductivity and permittivity. At any stage of its progress, let 
the medium become slightly electrically conducting all over, 
not merely in advance of the wave but behind it as well, for a 
reason that will presently appear. What happens to the wave 
now that the fresh influence is in operation ? 

A part of the answer we can give at once, by the pre- 
vious. The wave-sheet will move on just as before, but will 
attenuate as it goes, according to the time- factor e~ kt i 2c . Since 
we suppose the conductivity to be slight, it follows that a great 
distance may be traversed before there is notable attenuation. 
We also know that the total induction remains constant. 
The rest of it that is, what is not in the sheet at any moment 
is therefore left behind. The rejecting process commences the 
moment the conductivity is introduced, and continues to act 
until the plane wave is attenuated to nothing. The rejected 
portions travel backwards. But they are themselves subject 
to the same laws as the main plane wave, and so get mixed up. 
The result is that at time t after the introduction of the con- 
ductivity, the whole region of disturbance extends over the 
distance 2vt, half to the right and half to the left of the 
initial position. At the advancing right end we have a strong 
condensed disturbance, viz., the original wave attenuated, and 
behind it a weak diffused one. We can therefore, without 



misunderstanding refer to them as the head and the tail, with- 
out any body to complicate matters. Now the nature of the 
tail is quite different as regards the displacement and the 
induction. It is therefore convenient to regard one of them 
alone in the first place, and, of course, we select the induction, 
on account of the simple property of persistence that it 
possesses. We can distinguish three or four different stages in 
its development. 

The first stage is when the attenuation of the head is not 
great, say, whilst the head decreases from 1 to - 75. Whilst 
this occurs, the total induction in the tail rises from to 0'25. 
The tail is long and thin, and tapers to a point at its extreme 
end, or tip, at distance 2vt behind the head, and is thickest 
where it joins on to the head. 

The second stage roughly belongs to the period during which 
the head further attenuates from 0'75 to 0'5 or 0'4. The total 
induction in the tail then increases from 0"25 to 0-5 or 0'6. 
During this stage we find that the tail, which has, of course, 
greatly increased in length, does not go on increasing in thick- 
ness at the place where it is developed, but stops increasing 
and shows a maximum at or near that place. 

The third stage occurs during the further attenuation of the 
head to, say, 0*1, whilst the total induction in the tail increases to 
0*9. The maximum thickness of the tail is now a long way from 
the head, and at the end of the stage is nearer to the middle than 
to the head. Of course, since the head itself is now so small, 
the additions made to the tail must also become smaller. 

The fourth stage is when the head practically disappears 
and all the induction is in the tail. The maximum thickness 
is now nearly in the middle on the right side, however and 
the tail is nearly symmetrical with respect to its middle, where 
there is a swelling, beyond which the tail tapers off both ways 
to its two tips.* 

The final state is the consummation of the previous, and is 
one of perfect symmetry with respect to the middle of the tail, 
which is situated exactly where the plane wave was when the 

* As the division into distinct stages is somewhat arbitrary, this descrip- 
tion of the transition from an elastic to a diffusion wave should be under- 
stood to be only roughly approximate. It is made up, not from the 
formula, but by a numerical process of mixture. 


spreading began. The spreading now takes place according to 
the pure diffusion law, as of heat by conduction. 

Now as regards the displacement in the head and tail in the 
different stages. In the first and second stages the displace- 
ment is wholly negative in the tail, assuming it to be positive 
in the head, where, it should be remembered, it attenuates in 
the same manner as the induction which accompanies it. Thus, 
when the head has fallen to 0'9, the total displacement in the 
head and tail has fallen to (0'9) 2 or 0-81, so that the total nega- 
tive displacement in the tail is of amount 0'09, which is not 
much less than the coincident induction. And when the head 
has attenuated to 0*8, and the total displacement to (0*8) 2 or 
0-64, the negative displacement in the tail amounts to 0'16. 
But, unlike the induction, the displacement increases in the 
tail from the head up to not far from the tip, where, of course, 
it falls to zero. There is no tip at the forward end. But as 
the tail stretches out further to the left, and has fresh addi- 
tions made to it on the right side, the decrease of the density 
of displacement in passing towards the head continues, until 
somewhere about the end of the second stage, it becomes zero 
next the head. This node is approximately at the place where 
the induction has its maximum. When the head has fallen to 
0*4, we have the total displacement attenuated to 0*16, so that 
the negative displacement in the tail amounts to 0'24. 

In the third stage the displacement is negative from the tip 
up to somewhere near and beyond the maximum of induction, 
and increasingly positive in the remainder, up to the head. 
That is, the region of positive displacement now extends itself 
from the head a good way into the tail. At the same time the 
place of maximum negative displacement moves forward. 

During the fourth stage, the place of maximum negative 
displacement shifts itself to nearly the middle of the region 
between the tip and the node, beyond which the positive dis- 
placement has a nearly similar distribution, with a maximum. 
But this positive distribution is only three-parts formed, as the 
bead is still of some importance. The fifth stage completes 
the formation of the tail, with the displacement negative in 
one half and positive in the other, and nearly symmetrical 
with respect to the middle. Finally, we come to a state of 
perfect symmetry, with one maximum of induction and two 



(or a maximum and a minimum) of displacement. This may 
be readily understood by considering that in the expression for 
the true electric current, &E + cE, the second term finally be- 
comes a small fraction of the first, or the current is practically 
the conduction-current only. Then B, and therefore the dis- 
placement, is proportional to the space-variation of the induc- 
tion. Owing to the tail being made up of a complicated mix- 
ture of infinitesimal electromagnetic waves going both ways, 
we lose sight of the fundamental property of elastic wave-pro- 
pagation in a dielectric, the resultant effect being propagation 
by diffusion now, or very nearly so. The approximation to this 
result is closest in the middle of the tail. At the tips, on the 
other hand, we still have elastic waves, but, of course, of 
insensible strength. 

Now, suppose that the initial state is one of induction only, 
though still in a plane sheet. It may be regarded as the coin- 
cidence of two plane electromagnetic sheets of half the strength, 
with similar inductions and opposite displacements. The sub 
sequent history may, therefore, be deduced from the preceding. 
The initial induction splits into two halves, of which one moves 
to the right at speed v, and the other to the left. The history 
of the first wave as distorted by the conductance has been 
given. That of the second wave is the same, if we allow for 
the changed direction of motion and sense of the displacement. 
So we have only to superpose the two systems to show how an 
initial distribution of induction in a plane sheet splits and 
spreads, and the accompanying electric displacement. We 
have two equal heads, separating from one another at speed 2v, 
whilst the two tails unite to make a stouter kind of tail 
(referring to the induction) joining the two heads. This tail 
is always thickest in the middle. In fact, the distributions of 
induction and displacement are symmetrical with respect to 
the initial position from the first moment. In the final state 
that is tended to, when the tails have vanished, the induction 
is distributed in the same way as in the former case, in spite 
of the remarkable difference in the initial phenomena. 

If, again, the initial state is one of displacement alone in a 
plane sheet, this generates two oppositely-travelling electro- 
magnetic waves in which the displacements are similar and the 
inductions are opposite. The result is therefore to be got by 


combining the previous solution for a wave of positive induc- 
tion and displacement moving to the right, with a similar wave 
of negative induction and positive displacement moving to the 

Whatever be the magnitude of the conductivity, if finite, the 
same phenomena occur during the conversion of elastic waves 
to diffusion waves, and they may be represented by the same dia- 
grams, if suitably altered in the relative scale of ordinates and 
abscissae. But by increasing the conductivity from the small 
value which makes the above-described process take place over 
a considerable interval of time, we make it occur in a small 
interval only; and when we come to what are usually con- 
sidered good conductors, viz., metals, then the interval of time 
is so small that we may say practically that the heads vanish 
almost at 6nce, and before the propagation has proceeded any 
notable distance. The rest of the story is the spreading by 
diffusion or mixture. Thus we have an important practical 
distinction between the very good and the very bad conductor, 
as regards the manner of propagation of induction. In the 
former the electric displacement is of no account hardly, 
except perhaps for very short waves, and the practical theory 
is the theory of diffusion. At the other extreme are perfect 
non-conductors, in which the propagation is entirely by elastic 
waves. Between the two we have, in bad conductors, a mix- 
ture of the two kinds, though with a continuous transition 
from one kind to the other. The mathematical treatment of 
elastic waves is the easiest. The next in order of difficulty are 
the waves of pure diffusion, by the ordinary Fourier mathe- 
matics. The most troublesome are the intermediate forms of 
changing type, and the full analytical results are best got by 
a generalised calculus. But the general nature of the results 
may be obtained approximately by easy numerical calculations, 
with diagrammatical assistance. This is explainable without 
difficulty, and may be entered upon later on, when we come to 
detailed problems. 

la the above the conductivity has been electric only, and 
we have seen that there is a great difference between its effects 
on the electric and on the magnetic flux, which arises from the 
positive reflection and persistence of the latter, and the nega- 
tive reflection and subsidence of the former flux. The corres- 


ponding effects in a magnetic conductor may be readily 
deduced by the proper transformations, and exhibit analogous 
differences. Thus, starting with a pure plane wave sheet 
moving to the right, it is now the displacement that is con- 
served and divided between the head and tail, being initially all 
in the head, and finally all in the tail. On the other hand, the 
induction is negative in the tail at first and until the attenua- 
tion of the head is considerable. After this, the region of posi- 
tive induction extends from the head into the tail. As time 
goes on, and the head disappears, we have negative induction 
in one half and positive in the other (forward) half, whilst the 
displacement is positive all along it, with its maximum nearly 
in the middle. In the final state of diffusion we have symmetry 
with respect to the initial position of the wave, and the induc- 
tion is proportional to the space-variation of the displacement. 
The second term of the magnetic current ^H + juH is then a 
small fraction of the first term. 

Now, in the medium with duplex conductivity we must take 
the distortionless condition for the standard state. When this 
obtains there is no tailing, although the head (displacement and 
induction together) attenuates according to the time-factor c~^/ c . 
But if tho electric conductivity be in excess (that is, greater than 
is required to make k/c=g/fjL) there is tailing of the kind de- 
scribed as due to electric conductivity, with the induction posi- 
tive and the displacement negative in the tail at first. And if 
the magnetic conductivity is in excess, the tailing is of the 
other kind, with positive displacement and negative induction 
at first. But of course the details are not the same, because 
both the fluxes now attenuate to zero in time, in the tail as well 
as in the head. 

Application to Waves along Straight Wires. 

199. We must now endeavour to give a general idea of 
the tailing of waves when they run along conducting wires, in 
co-ordination with the previous. In the first place, observe 
that although the lines of electric and magnetic force in a pure 
electromagnetic sheet in ether must be always perpendicular to 
one another, yet they need not be straight lines, except in their 
elementary parts. On the contrary, we may have an infinite 


variety of distributions of the electric and magnetic fluxes in 
curved lines in sheets which shall behave as electromagnetic 
waves. Keeping to plane sheets, we may take any distribution 
of displacement in a sheet that we please, and make it an 
electromagnetic wave by introducing the appropriate distribu- 
tion of induction, also in the sheet. If left to itself it will 
move through the ether one way or the other at speed v, 
according to the directions of the fluxes. It should be noted, 
however, that the fluxes should have circuital distributions. 
For if not, and there is electrification or its magnetic analogue, 
they, too, should be moved through the ether so as to exactly 
keep up with the wave. If the electrification does not move 
thus, we have a changed state of things. But there is a way 
out of this difficulty. Let the displacement, when it is discon- 
tinuous, terminate upon infinitely-conducting lines, or surfaces 
of cylinders, according to circumstances. The interference 
produced by holding back the electrification will then be done 
away with. It is sufficient for explicitness to take a single 
definite case. 

Let the displacement in a plane electromagnetic sheet ter- 
minate perpendicularly upon a pair of perfectly-conducting 
cylindrical tubes placed parallel to one another. One is for the 
positive and the other for the negative electrification. The 
induction will then go between and round the tubes. One 
tube may enclose the other, but, preferably, we shall suppose 
this is not the case, so that our arrangement resembles a pair 
of parallel wires. Now the wave will run through the ether 
in the normal manner, without distortion or change of type, 
and will carry the electrification (on the conductors) along 
with it. 

Similarly if the tubes be magnetic conductors, if the induc- 
tion terminates perpendicularly upon them, whilst the displace- 
ment goes between and round them. So far, then, the theory 
of the propagation of the wave is unaltered. 

But this property may be greatly extended. The medium 
outside the tubes through which the wave is moving may be 
made electrically conducting. The theory is then identical 
with that of 198, as regards the distortion produced and the 
gradual destruction of the wave, ending in the process of diffu- 
sion. Thid will be true whether the tubes are perfect electric 


or magnetic conductors, provided we have the displacement 
terminating upon them in the first case and the induction in 
the second. 

Again, if the medium is magnetically conductive, the theory 
still holds good, with the same reservations as regards termi- 
nating the fluxes. But now the distortion is of the other kind, 
with persistence of the displacement and subsidence of the 

Finally, the theory is still true when the medium is a duplex 
conductor, whilst the tubes are either perfect electric or mag- 
netic conductors, according to choice. The distortion is now 
of the electric or of the magnetic kind, according as kfc is 
greater or less than g/fj.. 

But not one of these cases is quite what we want to repre- 
sent propagation along real wires. The nearest approach is 
the case of finite electric conductivity in the medium combined 
with perfect electric conductance of the tubes. If we make 
their conductance be imperfect, we then come close to the real 
problem. Now when we do this, the theory, when done pre- 
cisely, becomes excessively difficult, for two reasons. First, 
the waves in the ether outside the tubes are no longer plane ; 
and next, the penetration of the disturbances into the sub- 
stance of the tubes in time by means of cylindrical waves has 
to be allowed for. The latter is, of course, more necessary 
when solid wires are in question. A practical working theory 
is seemingly impossible of attainment by strictly adhering to the 
actual conditions. But one is possible by taking advantage of 
the fact that the waves in the ether are very nearly plane under 
ordinary circumstances. Of this we may assure ourselves 
by considering that the tangential component of the electric 
force at the surface of a wire (upon which the penetration 
depends) is usually a very small fraction of the normal com- 
ponent of the same outside it. The practical course, then, is 
to treat the waves in the dielectric as if they were quite plane. 
This does not prevent our allowing for the distorting effect of 
the resistance of the wires. It has the effect of making our 
solutions approximate instead of complete. But the important 
thing is to have a theory that, whilst sufficiently accurate, is 
practically workable, and harmonises with more rudimentary 
theories. This is precisely v;hat we do get, as done by me in 


1886. The theory is brought to such a form that we may 
employ it in several ways, according as circumstances allow us 
to ignore this or that influence. We may, for example, treat 
the wires as mere resistances (constant), and this is quite suffi- 
cient in a variety of applications. Or, using the same equations 
with more general meanings attached to the symbols, we may 
find the effects due to the imperfect penetration of the mag- 
netic induction and electric current into the wires when sub- 
jected to varying forces at their boundaries, either simple 
harmonically or otherwise. Furthermore, the theory is in such 
a form that it admits readily and without change of the intro- 
duction of terminal or intermediate conditions of the kind that 
occur in practice, whose effects are brought in according to the 
usual equations of voltage and current in arrangements of 

What we are immediately concerned with here, however, is 
the connection between this theory and the general theory of 
waves in a medium of duplex conductivity. When the tubes 
are so thin that penetration is practically instantaneous (for 
the wave-length concerned), a constant ratio is fixed (assisted 
by Ohm's law) between the intensities of magnetic and of tan- 
gential electric force at the boundary of the tube, where it 
meets the dielectric. And when we incorporate this result in 
the second circuital law applied to any section of the circuit 
formed by the parallel tubes, we find that the result is to turn 
it to the form expressing the existence of magnetic conductivity 
in the outer medium, without resistance in the wires. That is, 
the resistance of the wires has the same effect in distorting 
and dissipating the waves outside as the fictitious magnetic 
conductivity in the medium generally. This is true in the 
first and most important approximation to the complete theory. 
It is a point that is not altogether easy to understand, because 
the magnetic conductivity is fictitious, whereas the resistance 
is real. This, however, may be noted, that the theory of 
propagation of plane waves in a medium of duplex conduc- 
tivity bounded by perfect conductors for slipping purposes, 
professes to be a precise theory, whereas the other, although 
concerning the same problem in most respects, professes to be 
only approximate so far as the influence of the wires is con- 


Transformation of Variables from Electric and Magnetic 
Force to Voltage and Gaussage. 

200. In the consideration of the transmission of waves 
along wires, especially, of course, in practical calculations, it is 
more convenient to employ the line-integrals of the electric and 
magnetic forces as variables rather than these forces them- 
selves. This transformation of variables involves other neces- 
sary transformations, and the connection between the old 
quantities, suitable for waves in general, and the new, should 
be thoroughly understood, if something more than a super- 
ficial knowledge of the subject be desired. 

Thus, commencing with a circuit consisting of a pair of 
parallel straight conducting tubes, of no resistance in the first 
place, take for reference any plane which crosses the tubes per- 
pendicularly. It is the plane of the wave at the place, and the 
lines of electric and magnetic force lie in it, the former starting 
from one (the positive) and ending upon the other (the negative) 
tube, the latter passing between and round them. These lines 
always cross one another perpendicularly, but their distribution 
in the plane may be varied by changing the size and form of 
section of the tubes by the plane and the distance of separation. 
Now there is no axial magnetic induction that is, induction 
parallel to the tubes. Nor is there any tangential component 
of electric force on the surface of the tubes. It follows, by the 
second circuital law, that the line-integral of the electric force 
from one tube to the other is the same by any path in the 
reference plane. It is not the same by any path with the 
same terminations if we depart from the reference plane, but 
that is not yet in question. Call this constant line-integral V. 
It is the transverse voltage. In another form, we may say 
that the circuitation of the electric force in the reference plane 
is zero. 

Next consider the magnetic force. Its circuitation is also 
zero in the reference plane, provided the one tube or the other 
is not embraced, because there is no axial electric current in 
the dielectric. But when a tube is embraced the circuitation 
is finite. Call this quantity C. It is " the current " in the 
tubes, or, at any rate, is the measure thereof, and is positive 
for one tube and negative for the other. These two quantities, 


V and C, the transverse voltage and the current, are the new 
variables. They are as definite as E and H themselves in a 
given part of the reference plane under the circumstances. 

Next, as regards the fluxes, displacement and induction. Intro- 
duce a second reference plane parallel to the first and at unit 
distance in advance of it along the tubes. Anywhere between 
the planes we have D = cE, where E and D vary according to 
position. Now, as we substitute the complete voltage for E, 
so we should substitute the complete displacement for D. The 
complete displacement is the whole amount leaving the posi- 
tive and ending upon the negative tube, between the reference 
planes. That is, it is the " charge " per unit length of the 
tubes, say Q. As D is a constant multiple of E, so is Q a con- 
stant multiple of V. Say Q = SV. This S is the permittance 
of the dielectric, per unit length axially. It is proportional to 
c, the permittivity, but of course involves the geometrical data 
(brought in by the tubes) as well. 

Similarly, we have B = /xH everywhere between the reference 
planes, H and B varying from place to place. But if we sub- 
stitute the complete circuitation of H, we should also substi- 
tute the complete integral of B. This means the total flux of 
induction passing between and round the tubes, between the 
reference planes. Call it P. It is the magnetic momentum 
per unit length axially. As B is a constant multiple of H, so is 
P a constant multiple of C. Thus P = LC, where L is the 
inductance per unit length axially. This L varies as the induc- 
tivity /*, and involves the geometrical data. 

In virtue of the relation /*cv 2 = 1, the inductance and permit- 
tance are reciprocally related. Thus, LSv 2 = 1. Since v is con- 
stant, it might appear that the inductance was merely the 
reciprocal of the permittance, or the elastance, with a constant 
multiplier, only to be changed when the dielectric is changed. 
But there is much more in it than this. Different physical 
ideas and effects are conditioned by inductance and permit- 
tance. Inductance and inductivity involve inertia, whilst 
permittance and permittivity involve compliancy or elastic 

Observe that P and Q are analogous, being the total mag- 
netic and electric fluxes, whilst V and C are also analogous, the 
voltage and gaussage respectively. So the ratios L and S axe 


also strictly analogous, in spite of their quantitative recipro- 
cality. They are both made up similarly to conductance, only 
in the case of permittance it is reckoned across the dielectric 
from tube to tube, and in the case of inductance round and be- 
tween the tubes that is, the magnetic circuit is closed, and 
the electric circuit unclosed. There are other ways of setting 
out the connections, but the above way brings out the analogies 
between the electric and magnetic sides in a complete manner. 
The quantity C may be quite differently regarded when the 
magnetic field penetrates into the substance of a tube (then to 
be of finite conductance), viz., as the total flux of conduction 
current, or sometimes of displacement current as well, in a 
tube. But the above method is independent of the penetration, 
holding good whether there is penetration or not, saving small 
corrections due to the waves not being quite plane. The mag- 
netic view of C, as the gaussage, is also the nearest to expe- 
rimental electrical knowledge. The other view is darker, 
because we cannot really go inside metals and observe what 
is going on there, or the forces in action. They must be 
inferential in a greater degree than the external actions. 

The electric energy per unit volume being JED or JrE 2 , 
when this is summed up throughout the whole slice of the 
medium contained between the reference planes at unit dis- 
tance apart, we obtain the amount JVQ or JSV 2 , which is 
therefore the electric energy per unit of length axially. 

Similarly, the magnetic energy density JHB or J/*H 2 , when 
summed up throughout the slice, amounts to JCP or JLC 2 . 
The reader should, in all these transformations, compare the 
transformed expressions with the original, and note the proper 
correspondences. It is all in rational units, of course, to avoid 
that unmitigated nuisance, the 4?r factor of the present B.A. 

The density of the flux of energy, which is the vector pro- 
duct of E and H in general, is of amount simply EH when 
the forces are perpendicular, as at present. When this is 
summed up over the reference plane, the result is VC, the pro- 
duct of the voltage and current. The flux of energy is parallel 
to the guiding tubes, and EH is the amount for a tube in 
the dielectric of unit section, whilst VC is the total flux of 


The relation E = p>H, or, which means the same, H = cvE, 
which obtains in a pure electromagnetic wave, becomes converted 
to V = LvC, or C = SvV. We see that Lv is of the dimensions of 
electric resistance, and Sv of electric conductance. Similarly, 
because V and C are the line-integrals of E and H, we see that 
fjiv is of the dimensions of resistance, and cv of conductance. 
The activity product VC is what engineers have a good deal to 
do with, now that " electrical energy," or energy which has been 
conveyed by electromagnetic means, is a marketable commodity. 
However mysterious energy (and its flux) may be in some of its 
theoretical aspects, there must be something in it, because it is 
convertible into dollars, the ultimate official measure of value.* 

Transformation of the Circuital Equations to the Forms 
involving Voltage and Gaussage. 

201. Now, still under the limitation that the guiding tubes 
have no resistance, and that the dielectric has no conductance, 
consider the special forms assumed by the circuital laws. These 
are, in general, 

curlH=cE, .,,,.. (1) 


Now, in applying these to our plane waves, we may observe 
at the beginning that E and H and their time- variations are in 
the reference plane, and have no axial components. So the 
only variations concerned in the operator " curl " are such as 
occur axially, or parallel to the guides, f Let x be distance 
along them, then the circuital laws become 

-^? = cE, (3) 

dx v ' 

~ = ^ (4) 

dx x ' 

* See the Presidential Address to the Institution of Electrical Engineers, 
January 26, 1893. 

t That is, curl reduces from Vv, where y is i\?i +JV^ + kVs in general, to 
% r iVi simply. It is now more convenient to use the tensors E and H, as 
in (3) and (4), ignoring their vectoiial relations. 


which apply anywhere in the reference plane. Transforming 
to V and C, we obtain 


By (3) and (4) we see that the space-variation of H is the 
electric-current density, and the space- variation of E the mag- 
netic-current density in the dielectric. By (5) and (6) we 
express the same truths for the totals concerned. 

We may see the meaning of (5) thus. Draw a closed line in 
the reference plane on and embracing the positive guiding tube. 
The circuitation of H there is the total surface current C. Let 
the closed line be shifted to the next reference plane at unit 
distance forward. Then dC/dx is the amount by which C 
decreases during the shift, and (5) tells us that it is equiva- 
lently represented by the time-rate of increase of the charge on 
the tube between the reference planes, or the rate of increase 
of the total displacement outward. This process may be applied 
to any closed line in the reference plane, provided it embraces 
the tube. When it is shifted bodily forward through unit 
distance to the next reference plane, the amount by which the 
circuitation of H, that is, C, decreases from the first to the 
second position measures the displacement current through 
the strip of the cylinder swept out by the closed line. This 
is the time-rate of increase of the charge per unit length of 
the tubes, and is the total transverse current from one tube 
to the other. 

The meaning of (6) requires a somewhat different, though 
analogous, elucidation. Join the tubes by any line in the first 
reference plane. The line-integral of E along it is V, the 
transverse voltage. Shift the line bodily along the tubes to 
the next reference plane. The transverse voltage becomes 
V + dV/dx, still reckoned from the positive tube to the 
negative. But if we join the two starting points on the 
positive tube together, and likewise the ending points on the 
negative tube, we make a complete circuit, having two trans- 
verse sides and two axial sides. Now reckon up the voltage in 
this circuit. The axial portions contribute nothing, because 


the electric force is perpendicular to them. The voltage 
required is, therefore, simply the difference in the values 
of the transverse voltage in the other two sides, or dNjdx. 
By the second circuital law, it is also measured by the rate 
of decrease of induction through the circuit, that is, by -LC. 
Whence follows equation (6). 

The reader who is acquainted with the (at present) more 
"classical" method of treating the electromagnetic field in 
terms of the vector and scalar potentials cannot fail to be 
impressed by the difference of procedure and of ideas involved. 
In the present method we are, from first to last, in contact with 
those quantities which are believed to have physical signi- 
ficance (instead of with mathematical functions of an essentially 
indeterminate nature), and also with the laws connecting them 
in their simplest form. Notice that V is not the difference of 
potential in general. It sometimes degenerates to difference of 
potential, viz., in a perfectly steady state. But when the state 
changes, we cannot express matters in terms of an electric 

Now, still keeping the guiding tubes perfectly conducting, 
let the medium in which they are immersed be slightly con- 
ducting. The electric-current density, which was cE before, now 
becomes &E + cE, where the additional &E is the conduction- 
current density. Along with this there is waste of energy at 
the rate E 2 per unit volume. This waste of energy may also 
be regarded as a storage of energy, viz., as heat in the medium. 
But as it is not recoverable by the same means (reversed) which 
stored it, it is virtually wasted, and we have no further concern 
with it. 

We have next to consider the total waste in the slice of the 
medium between the two reference planes, in terms of the trans- 
verse voltage. It sums up to KV 2 , where K is the transverse 
conductance of the medium per unit length axially. This is 
proportional to the conductivity k, and involves geometrical 
data in the same way as the transverse permittance, as may be 
readily seen without symbolical proof, on considering that the 
conduction-currrent lines and the displacement lines are simi- 
larly distributed, whilst both are controlled by the transverse 
voltage. In fact, the conduction-current density &E sums up 


to KV, the complete transverse conduction-current, in the same 
way as the displacement density cE sums up to the total trans- 
verse displacement SV, in terms of the permittance and 

The first circuital law (1) becomes changed to 


by the addition of the conduction current. And its modified 
form (3) for our plane waves becomes 


whilst in terms of the new variables, voltage and gaussage, we 
have this extended form of the equation (5), 


Next, let the medium become magnetically conductive as 
well, no other change being made. The conduction-current 
density is ^H, and its distribution resembles that of the induc- 
tion itself, whilst its amount is controlled by the quantity G, 
the gaussage. The total magnetic-conduction current may 
therefore, be represented by KG, where E, is the magnetic 
conductance, which varies as the magnetic conductivity g, and 
involves the geometrical data in the same manner as the 
inductance does. 

Similarly, the rate of waste of energy due to the magnetic 
conductivity is 0rH 2 per unit volume, and the total rate of 
waste in the slice of the medium between the two reference 
planes amounts to RC 2 correspondingly. 

Finally, the effect of the magnetic conductivity is to turn 
the second circuital law from the elementary form (2) to 

-curlE = #H + /iH, .... (10) 

and, correspondingly, the special form (4) for our plane waves 
becomes turned to 

..... (11) 


This, in terms of V and C, is equivalent to 

-4^ = RC + LC, ...... (12) 


which is the proper companion to the equation (9) expressing 
the first circuital law. These equations (9) and (12) are the 
practical working equations. 

We have already remarked that the structure of the electric 
permittance and conductance are similar. From this it follows 
that the time-constant c/k is identically represented by S/K. 
Similarly, the time- constant njg is identical with L/R. 

The density of the flux of energy is, with the two conduc- 
tivities, still represented by the vector product of E and H, and 
therefore the total flux is still the activity product VC. To 
corroborate this statement, and at the same time show the 
dynamical consistency of the system, consider that if the quan- 
tity VC is really the flux of energy across a reference plane, the 
excess of its value at one plane over that at a second plane 
further on at the same moment represents the rate of storage 
of energy between the planes. Therefore, the rate of decrease 
of VC with x is the rate of storage between two reference planes 
at unit distance apart. Now, 

_i(VC)=-vf-cf ..... (13) 
dx dx dx 

On the right side use the circuital equations (9) and (12), and 
it becomes . . 

V(KV + SV) + C(RC + LC), . . . (14) 

or, which is the same, 

. . (15) 

By the previous, the first term is the total electric waste, the 
second is the total magnetic waste, the third is the increase of 
total electric energy, and the fourth is the increase of total 
magnetic energy, per unit of time, between the two reference 
planes. This proves our proposition, with a reservation to be 
understood concerning the circuital indeterminateness of the 
flux of energy. There is, therefore, no indistinctness anywhere, 
nor inconsistency. We have now to show in what manner 
the above is affected by the resistance, &c., of the guides. 



The Second Circuital Equation for Wires in Terms of V and 
C when Penetration is Instantaneous. 

202. Let the parallel conducting tubes be of finite resist- 
ance. As a result, the external disturbance penetrates into 
them, and a waste of energy follows. As a further result, the 
external disturbance becomes modified. This occurs in a two- 
fold manner. By cumulative action, the nature of a wave sent 
along the leads may be profoundly modified as it progresses. 
Besides this, there is what we may term a local modification, 
whereby the wave at any place is no longer strictly a plane 
wave. It is merely with this effect that we are now concerned. 
The departure from planarity requires a considerably modified 
and much more difficult theory, no longer expressible in terms 
of V and C simply, in order to take it into account. But in 
the construction of a practical theory, we take advantage of the 
fact that the departure from planarity is slight, to which we 
have already referred towards the end of 199. A line of elec- 
tric force does not now start quite perpendicularly from the 
positive lead and end similarly on the negative lead. It has a 
slight inclination to the perpendicular, and therefore curves out 
of the reference plane to a small extent. In the dielectric 
itself, this peculiarity is of little importance. But the slight 
amount of tangentiality of the electric force at the surface of 
the leads, which is conditioned thereby, is of controlling import- 
ance as regards the leads themselves, and eventually through 
them, to the waves in the dielectric, by attenuating and altering 
the shape of waves (considered axially), as just referred to. We 
should now consider what form is assumed by the second cir- 
cuital law in terms of V and C, when we admit that there is 
tangential electric force on the surface of the leads, but on the 
assumption that the minor effects in the dielectric itself, which 
are associated with the presence of the tangential force, are of 
insensible influence. 

We may still regard V as the transverse voltage in the 
reference plane. Whether we go straight across from the 
positive to the negative lead, keeping in the reference plane, 
or leave it slightly in order to precisely follow a line of force in 
estimating the voltage, is of no moment, because the results 
differ so slightly. The quantity C is subject to a similar slight 


difference, according to the way it is reckoned. We may, per- 
haps, most conveniently reckon it as the circuitation of the 
magnetic force round either lead upon its surface and in the 
reference plane, because this way gives the exact value of the 
conduction current in the lead. But the circuitation in other 
paths in the reference plane will not give precisely the same 
value now that the waves are not quite plane. This difference 
we also ignore in the practical theory. In truly plane waves 
the electric current in the dielectric is wholly transverse. There 
is now really an axial component as well, but being a minute 
fraction of the transverse current, it is ignored. In short, we 
have to make believe that the waves are planar when consider- 
ing their propagation through the dielectric, whilst at the same 
time we take into account the departure from planarity in 
considering the influence of the leads and what occurs in them. 
It is unfortunate to have to refer to small corrections, as it 
confuses the statement of the vitally important matters. Let 
us, then, set them aside now. 

Construct the second circuital equation in the manner fol- 
lowed in 201, in elucidating the meaning of equation (6). 
Consider a rectangle consisting of two transverse sides in 
reference planes at unit distance apart, beginning upon the 
positive and ending upon the negative lead, and of two axial 
sides of unit length upon the leads themselves. Reckon up the 
voltage in this rectangle. The transverse sides give V and 
V + dV/dx. The axial sides give E x and E 2 say, if Ej is the 
tangential component in the direction of increasing x of the 
electric force at the boundary of the positive tube, and E 2 the 
same on the negative tube, but reckoned the other way. The 
complete voltage is then 

or + 1 + 2 . 


It is also equal to - LC, as in 101. (There is no magnetic 

conductance of the dielectric now.) So 

.... (16) 

is the form assumed by the second circuital law. 



Observe that we have made no specification of the nature of 
the leads, so that the equation possesses a high degree of 
generality. As regards the first circuital law, that is unchanged, 
being expressed by (5) when the dielectric is non-conductive, 
and by (9) when conductive, since the minor changes alluded 
to as regards V and C are not significant. 

Being general, the equation (16) needs to be specialised 
before it can be worked. If our electromagnetic variables are 
to continue to be V and C, we require to express E : and E 2 as 
functions of C. Fortunately this can be done, sometimes very 
simply, and at other times in a more complicated way. To 
take the simplest case, let the leads be tubes (or sheets) of so 
small depth that penetration is practically instantaneous as 
waves pass along them. Then E x is not only the boundary 
tangential electric force, but is also the axial electric force 
throughout the substance of the positive tube. Similarly as 
regards E 2 for the negative tube. It is true that, in virtue of 
the transverse electric current from tube to tube, there is also 
transverse current in the tubes, and, therefore, transverse 
electric force, but this is to be ignored, because it is a small 
fraction of the axial. The current density in the positive tube 
is therefore axial, of strength ^E x , if ^ is the conductivity 
of the material; and the total current in the tube is KjEj, 
where Kj is the axial conductance per unit length, or, which is 
the same, Ej/R^ if R x is the resistance per unit length. But 
this quantity is also the previously investigated quantity C, 
the circuitation of the magnetic force on the boundary of the 
tube. So we have the elementary relations 

E^R^, E 2 = R 2 C 2 , . . . (17) 

which are, be it observed, essentially the connections of the 
electric and magnetic forces at the boundaries, though brought 
to a particularly simple form by the instantaneous penetration. 
The second circuital equation (16) therefore takes the form 


Or, finally, if R is the resistance per unit length of the two 

-^ = RC + LC, ...... (19) 



which is the practical equation for most purposes. Observe, 
comparing it with (12), that there is an identity of form, but 
with a changed meaning of the symbol R. In the exactly stated 
problem of plane waves running through a medium of duplex 
conductivity bounded by perfect conductors, the quantity R 
is the external magnetic conductance per unit length axially. 
In the present approximately stated problem of very nearly 
plane waves running through an electrically conducting medium 
bounded by resisting wires, the quantity R is the resistance of 
the wires, per unit length axially. What we have to do, there- 
fore, in order to turn the real problem into one relating to 
strictly plane waves admitting of rigorous treatment, is to 
abolish the resistance of the leads and substitute equivalent 
magnetic conductance in the dielectric medium outside. The 
two materially different properties are nearly equivalent in 
their effects. Equations (13), (14), (15), still hold good, only 
RC 2 is now to be the rate of waste in the leads, instead of in 
the medium generally due to the (now suppressed) magnetic 
conductance. The flux of energy is now not quite parallel to the 
leads everywhere, but has a slight slant towards them. But 
the usual formula gives the waste correctly. The product 
EjHj of the tangential electric force E x and the magnetic force, 
say H 1 , at the boundary of the positive lead, is the rate of 
supply of energy to the lead per unit surface. Therefore, by 
circuitation, the product EjC is the rate of supply per unit 
length axially. This is the same as the previous RjC 2 . 

The equivalence of effect of magnetic conductance externally 
and of electric resistance in the leads, is undoubtedly a some- 
what mysterious matter, principally because it is hard to see 
(apart from the mathematics) why it should be so. As regards 
the above reasoning, however, it is essentially simple, and is a 
direct application of fundamental electrical principles. It may, 
therefore, cause no misgivings, except the doubt that may 
present itself whether the ignored small effects in the dielectric 
due to departure from planarity of the waves are really ignor- 
able. As a matter of fact, they are not always of insensible 
effect, and plenty of problems can be made up and worked out 
concerning tubes and wires which do not admit of the compa- 
ratively simple treatment permissible when V and C are the 
variables, and the results are exceedingly curious and interest- 


ing, though quite unlike those at present in question. But 
these are not practical problems, and have little bearing upon 
the question of the free propagation of waves along long parallel 
straight wires. They do not start as full grown plane waves 
from the source of disturbance, but they very soon fit them- 
selves on to the wires properly, and then follow the laws of 
plane waves pretty closely. 

Although thin tubes for leads were specially mentioned in 
connection with equation (19), yet any sort of leads will do 
provided the penetration be sufficiently rapid to be instan- 
taneous " within the meaning of the Act." It is obviously true 
for steady currents, when the inertial term disappears and B, is 
the steady resistance. And if the variations of current be not 
sufficiently rapid to cause a sensible departure from uniformity 
of distribution of current in the conducting wires, then, of 
course, (19) may still be safely used. At the same time, it 
should be mentioned that the inductance L should, under the 
circumstances, be increased by a (usually) small amount due 
to induction in the conductors themselves, as will be presently 
noted more closely. Furthermore, remember that no allow- 
ance has been made for the influence of parallel conductors, 
should there be any. The earth does not count if the leads be 
alike and equidistant from the ground, as its influence can be 
embodied in the values of L and S, the inductance and per- 

The Second Circuital Equation when Penetration is Not 
Instantaneous. Resistance Operators, and their Definite 

203. The next question is what to do when penetration is 
not instantaneous within the meaning of the Act. We should 
first go back to (16), which remains valid, and inquire whether 
the tangential electric forces cannot be expressed in terms of 
the current (or conversely) in some other way than by a linear 
relation. Suppose we say 

E^B^C, E 2 = R" 2 C. . . . (20) 

Is it possible to give a definite meaning to the symbol R," ? Is 
there a definite connection between C, regarded as a function 
of the time, and E x or E 2 also regarded in this way ? 


Imagine a wire to be free from current, and, therefore, elec- 
trically neutral. Now expose its boundary to tangential 
electric force, beginning at a certain moment, varying in some 
particular way with the time later, and then ceasing. The 
result is that C, the total current in the wire, will run through 
a particular sequence of values, and then finally cease. If we 
begin again with the applied force, and make it run through 
the same values in the same manner, we shall again obtain the 
same values of C at corresponding moments. So far, then, the 
connection is a definite one. Moreover, if we make the applied 
force run through the same values increased in a certain ratio, 
the same for all, then the current will have its previous values 
increased in the same ratio. But if we change the nature of 
the applied force as a function of the time (irrespective of size) 
we shall find that C is not merely changed as a function of the 
time, but also as a function of the applied force. That is, the 
mere value of one does not necessitate any particular simulta- 
neous value of the other. So, if we keep to the usual sense 
meant when algebraists say u =/(#), or u is a function of #, we 
cannot say that C is a function of E. It is, nevertheless, true 
that the march of C is strictly connected with that of E, so 
that when the latter is given, the former is obligatory. To 
deny this would be equivalent to the denial of there being 
definite controlling laws in operation. The full connection 
between E and C, however, involves not merely their values, 
but also the values of their first, second, third, <fec., differential 
efficients up to any order. That is, the symbol R", taken by 
-,self, is a function of the differentiator d/dt. To illustrate by 
simple example, suppose 

R" = R + L P + ($p)-\ .... (21) 

ere R, L, S, are constants, and p stands for the differentiator. 
's means that 

}0, .... (22) 

or E = RC + LC + S/Ccft, .... (23) 

in the common notation of integrals. Now imagine the march 
of C to be given. This implies that the march of C is also 
known, and likewise that of /Cdt. Consequently the march of 


E is explicitly known. It is not obvious that when the march 
of E is given, that of C is known, by the same operator impli- 
citly. That is, by (22) we have 


and given E as a function of the time, C is known as a function 
of the time, or if not known, can be found without ambiguity. 
It is not obvious, because we do not immediately see how the 
operation indicated in (24) is to be carried out, whereas, in the 
case of (22), it is visible by inspection. Nevertheless, the fact 
that the march of E, physically considered, conditions that of 
C, makes the above equation (24) not only definite, but com- 
plete. In the usual treatment of the theory of differential 
equations, there is no such definiteness. Arbitrary constants 
are brought in to any extent, to be afterwards got rid of. Now 
this is all very well in the general theory of differential 
equations, where arbitrary constants form a part of the theory 
itself, but for the practical purpose of representing and obtaining 
solutions of physical problems, the use of such arbitrary and 
roundabout methods (which are too often followed, especially 
by elementary writers) leads to a large amount of unnecessary 
work, tending to obscure the subject, without helping one on. 
It would not, perhaps, be going too far to say that such a 
misuse or inefficient application of analysis often makes rig- 

When we say that E = R"C, where IT is the resistance 
operator (so called because it reduces to the resistance in 
steady states), we assert a definite connection between E and 
C, so that when G is fully given as a function of the time, and 
the operations contained in R" are performed upon it, the func- 
tion E results, and similarly, when it is E that is given, then 
the inverse operator (R")" 1 (or the conductance operator) act- 
ing upon it will produce C. There may be an infinite number 
of differentiators in R", as p, p 2 , p*, and so on, where p n mean? 
d^/df 1 . But there is not a single arbitrary constant involved 
in E R"C, nor, indeed, anything arbitrary. 

Returning to the wires, the form of the resistance operator. 
as a function of p, depends upon the electrical and geometrical 


data. It has been determined for round wires and round tubes, 
and plane sheets. We may take 

. . . (25) 

as the form of the second circuital law, and make the determi- 
nation of the operators a matter of separate calculation. 

As already mentioned, when C is steady, R"j degenerates to 
R x and R" 2 to R 2 , the steady resistances (per unit length) of 
the wires, tubes, rods, or cylinders of any shape that may be 
employed. At the same time the inertial term disappears. 
Also, when C varies, it is sometimes sufficient to take into 
account only the first approximation to the form of the opera- 
tors. This is, for a solid round wire, 

...... (26) 

and similarly for R" 2 . This J/^ is the value of the steady 
inductance of the wire, ^ being its inductivity. When not 
solid, or not round, some other expression is required. Notice 
that this brings (25) to the elementary form (19), because 
the inductance of the wires may be included in L itself, which 
then becomes the complete steady inductance (per unit length 
axially), including that due to the dielectric and that due to 
the two wires. This usually means only a small increase in 
the value of L, unless the wires be of iron. 

Simply Periodic Waves Easily Treated in Case of Imperfect 

204. But besides the above simplification, there is an ex- 
ceedingly important general case in which a similar reduction 
takes place. This occurs when the sources of disturbance vary 
simple periodically with the time. Then the electric and mag- 
netic fluxes everywhere vary ultimately according to the same 
law with the same period, provided the relations of the fluxes to 
the forces are linear ; that is, when the conductivity, permit- 
tivity, and inductivity, are constants at any one place. Now, 
when this comes to pass, both E and C in the equation E = R"C 
vary simple periodically with the time. But when the sine or 
cosine is differentiated twice, the result is the same function 
negatived, and with a factor introduced. Thus, if the frequency 


is ?&/27r, so that sin (nt + 0) may be considered to be the time- 
factor in the expression for either E or C, we have the property 
d 2 /dt 2 = - n 2 . Put, therefore p 2 = - n 2 in the expressions for the 
resistance operators, and we reduce them to 

B'j-B'j + L'ip, ll" 2 = R' 2 + L>, . . (27) 

where R' and L' are functions of n\ They are, therefore, con- 
stants at a given frequency, and this is a very valuable property, 
as it is clear at once that we reduce the equation (25) to the 
simple form 

. (28) 

or, more briefly and clearly, 

_^I = R'C + L'C, .... (29) 

which is the same as (19), valid when the penetration is in- 
stantaneous, but with different values of the constants involved. 
The steady resistance R is replaced by R', the effective resist- 
ance (of both tubes per unit length) at the given frequency, 
and L the steady inductance (inclusive of the parts due to the 
wires) by L', the effective inductance. 

Owing to the reduction of the second circuital equation to 
the primitive form, we are enabled to express the propagation 
of simply periodic waves along wires by the same formulae, 
whether there be or be not imperfect penetration. We simply 
employ changed values of the constants, resistance and induct- 
ance, which may be independently calculated, or left to the 
imagination, should the calculation be impracticable. This is, 
when it can be effected, the best way of making extensions of 
theory. Do the work in such a way that harmony is produced 
with the more rudimentary results, and so that they will work 
together well, and the new appear as natural extensions of th* 
old. An appearance of far greater originality may, indeed, be 
produced by ignoring the form of the elementary results, but 
the results would be cumbrous, hard to understand, and un- 

The effect of increasing the frequency from zero to a high 
degree is to first lessen the penetration, and end in mere skin 


penetration. Consequently, the quantity L', the effective in- 
ductance, goes from the full steady value L + L^ + L'g, and 
finishes at L simply, the inductance of the dielectric. But the 
difference need not be great. In the case of suspended copper 
telephone wires the inductance of the dielectric is far larger 
than the rest, so that there is no important variation in the 
value of the inductance possible, nor would there be were the 
frequency increased up to that of Hertzian vibrations. But as 
regards the effective resistance, the case is different. As the 
distribution of current in a wire changes from that appropriate 
to the steady state, the resistance increases, and the increase is 
not always a negligible matter. If long-distance telephony 
were carried on along iron wires it would be a very important 
effect. But the Americans, who were the introducers of long- 
distance telephony, soon found that iron would not do, and that 
copper would. The reason of the failure is mainly the largely 
increased resistance of iron. In copper, on the other hand, it 
is an insensible effect at the lower limit of telephonic frequency 
of current waves, with the size of wire employed, and is not 
very important at a frequency three or four times as great. 
On the other hand, in the numerous experiments with very 
rapid vibrations of recent years, due to Hertz, Lodge, Tesla^ 
and many others, the increased resistance due to imperfect 
penetration becomes a very important matter, and is one of the 
controlling factors that should be constantly borne in mind. 

Long Waves and Short Waves. Identity of Speed of Free 
and Guided Waves. 

205. By replacing fictitious magnetic conductance of the 
medium outside the pair of leads by real electric resistance of 
the leads themselves, we have obtained the same results as 
regards the propagation of waves, subject to certain reserva- 
tions referring to the practical applicability of the theory. It 
is worth while noticing, in passing, a certain peculiarity show- 
ing roughly when we may expect the theory to be admissible, 
and when it should fail. We know that in the transmission of 
waves along perfectly conductive leads the waves are con- 
tinuously distorted if the medium be electrically conductive. 
Also, that by introducing magnetic conductivity into the 


medium we may reduce this distortion, and ultimately abolish 
it when the magnetic conductivity reaches a certain value. 
Now observe here that the correcting influence of the magnetic 
conductivity is exerted precisely where it is required, namely, 
in the body of the wave itself. The wave is acted upon in 
every part by two counteracting distorting influences, so that 
the correction is performed exactly ; in other words, we obtain 
an exact theory of distortionless transmission. 

But, on the other hand, when we employ the resistance of 
the leads to perform the same functions, the correcting in- 
fluence is not exerted uniformly throughout the body of the 
wave, but outside the wave altogether ; at its lateral boun- 
daries, in fact. Nevertheless, for reasons before stated, we still 
treat the waves as if they preserved their planarity under the 
influence of the resisting leads. Whilst, therefore, we fully 
recognise and employ the finite speed of propagation of disturb- 
ances axially, or parallel to the leads, we virtually assume that 
the correcting influence of the leads id transmitted laterally 
outwards instantaneously. We may, therefore, perceive that 
the wave-length is a matter of importance in determining the 
applicability of the practical theory. It is of no moment what- 
ever in the exact theory employing magnetic conductance ; but, 
when we remove this property from the medium generally, and 
(virtually) concentrate it at the leads, we should at the same 
time keep the wave-length a considerable multiple of the dis- 
tance between the leads if the practical theory is to be applic- 
able. To see this, it is sufficient to imagine the case of waves 
whose length is only a small fraction of the distance between 
the leads, when it is clear that their correcting influence could 
no longer be assumed to be exerted laterally and instantaneously, 
as if the small portion of the leads between two close reference 
planes belonged to and was associated solely with the slice of 
the medium between the same planes. 

We have referred in the above to the action of the resistance 
of the leads as a correcting one, neutralising the distortion due 
to another cause. But the same reasoning is applicable when the 
action is not of this nature. Thus, when the external medium 
is non-conductive, and the leads are non-resistive, we have 
perfect transmission. Making the leads resistive therefore now 
brings on distortion. We may now say that in order that this 


distortion should occur in the same way as under the influence 
of magnetic conductivity in the external medium, the wave- 
length should not be too small, as specified above. The prac- 
tical theory is therefore the theory of long waves, and in the 
interpretation of the word "long," some judgment may be 
exercised as regards the leads and other matters, because there 
is no hard and fast distinction, and what may be a long wave 
under some circumstances may be a short one under others. 

To illustrate this point, imagine that we have got an inge- 
nious instrument (not yet made), for continuously recording 
the electromagnetic state of a non-conducting medium, say the 
air at a certain place (just as we have thermometric and 
barometric recorders), and that this instrument is so immensely 
quick in its action as to take cognisance of changes happen- 
ing in very short intervals of time, say one thousand-millionth 
of a second. Now in applying this instrument to register the 
state of air traversed by electromagnetic waves, it is clear that 
the size of the waves must be considered in relation to the size of 
the instrument. If the instrument were one decimetre across, 
then waves of one kilometre in length could as well be regarded 
as of infinite length. But if only one metre in length, though 
it could still be used, there would be no longer the same accu- 
racy of application. And if of only a centimetre in length, 
then it is plain that several waves would be acting at once on 
the apparatus in different parts, and the resultant effect 
recorded would not represent the history of the waves by any 

Now as regards waves sent along parallel leads, it is obvious 
that light waves are totally out of the question, being im- 
mensely too short. On the other hand, telephonic waves are to 
be treated as long waves very long, in fact though they are 
short compared with telegraphic waves. But it is somewhat 
curious that the electromagnetic waves investigated by Hertz are 
sometimes so short as to come within the scope of the above 
reservatioDal remarks, or of others of a similar nature. Whether 
the generation of waves by an oscillator be considered, or their 
effect on a resonator, or their transmission along leads, their 
shortness in relation to the apparatus employed may some- 
times vitiate the results of approximate theories, and render 
caution nec r ssary. For example, the plane-wave theory indi- 


cates that the attenuation factor for waves running along 
parallel leads is e- Ra; / 2Lt> in the distance x, or c- 1 "/ 21 ' in the 
equivalent time of transit t. This is when R and L are con- 
stants. If not, and the waves be simply periodic, then we may 
use the same formula with the effective values of R and L at 
the frequency concerned. But if this be true for long waves, 
we cannot expect it to continue true on shortening the waves 
to the transverse distance of the leads, more especially if we 
are ignorant of the precise type of the waves. At the best, we 
should not expect more than results of a similar kind. But 
not much has been done yet in the quantitative examination of 
Hertzian waves, for sufficiently obvious reasons. 

In one respect, however, a formerly very strange anomaly 
has been cleared up satisfactorily. When Hertz opened people's 
eyes and made them see the reality of Maxwell's ether as a 
medium propagating electromagnetic disturbances at the speed 
of light, by showing their transmission across a room and 
reflection by a metallic screen, the full acceptation of Maxwell's 
theory was considerably hindered for a time by his finding 
that the speed of waves sent along wires was much less than 
that of free waves. The discrepancy was a large one, 
and gave support apparently to the old view regarding 
the function of wires, which made the wires the primary 
seat of transmission, and effects outside secondary, due 
to the wires. And it came to pass that people, whilst 
admitting the truth of Maxwell's theory, yet made a distinc- 
tion between waves in free space and " in wires." This was 
thoroughly out of harmony with Maxwell's theory, which 
makes out that the wires, though of great importance as 
guides, are nevertheless only secondary. On the other hand, 
it should be mentioned that Lodge found no such large de- 
parture from the speed of light in his experiments. But the 
matter has been explained by the discovery that an erroneous 
estimate was made of the permittance of the oscillator in the 
experiments which apparently showed that the speed was largely 
reduced. When corrected, there is not left any notable differ- 
ence between the speed of a free wave and of one guided by 
wires. Of course, there is no reason why reference to waves 
" in wires " should not be dropped, unless the laterally-propa- 
gated cylindrical waves are meant. These are secondary, and 


have no essential connection with the propagation of the 
primary waves through the external dielectric, although modi- 
fying their nature. 

The Guidance of Waves. Usually Two Guides. One suf- 
ficient, though with Loss. Possibility of Guidance 
within a Single Tube. 

206. When waves are left to themselves in ether without 
the presence of conductors, they expand and dissipate them- 
selves. Even if they are initially so constituted as to converge 
to centres or axes, they will subsequently expand and dissipate. 
To prevent this we require conducting guides or leads. Now 
this usually involves dissipation in the leads ; but the point at 
present under notice is the property of guidance only. We 
can stop the expansion in a great measure, and cause a wave 
to travel along wherever we wish it to go. Practically there are 
two leads, as a pair of parallel wires ; or if but one wire be 
used, there is the earth, or something equivalent, to make 
another. But it is still much the same, as regards guidance, 
when there is but one wire, if we choose to imagine the case of 
a single infinitely-long straight wire alone by itself in ether. 
If we make it the core of a plane electromagnetic sheet, this 
sheet will run along the wire just as well and in the same way 
as if there were a second guide. But the energy of such a 
sheet, even though of finite depth, and containing electric and 
magnetic forces of finite intensity, would be infinite. The 
quantity L, the inductance per unit length of guide, is infinite 
under the circumstances. We could not, therefore, set up such 
a wave from a finite local source. If we cause an impressed 
voltage to act axially for a very short interval of time across 
any section of the guide, say in a reference plane, the result is 
an approximately spherical wave. (To be perfectly spherical, 
the wire should be infinitely fine. The case is then that of a 
spherical wave-sheet with conical boundaries, already referred 
to, with the angle of the cone made infinitely small.) Its 
centre is at the origin of the wave, and as it expands, the por- 
tions of the wave-sheet nearest the wire become approximately 
parallel plane waves, one going to the right, the other to the 
left along the wire. But not being pure plane waves they are 
weakened as they progress, by the continuous expansion of the 


spherical wave of which they form a part. Nevertheless, we 
have the propagation of nearly plane waves of finite energy, or 
of a perfectly plane wave-sheet of infinite energy, along a single 

Now, this takes place outside the conducting guide, and the 
question presents itself whether we cannot transmit an electro- 
magnetic wave along the interior of a tube, in a manner resem- 
bling a beam of light ? We can certainly do so if we have a 
second guide within the first tubular one, for this does not 
differ substantially from the case of two parallel wires, each out- 
side the other. But it does not seem possible to do without 
the inner conductor, for when it is taken away we have nothing 
left upon which the tubes of displacement can terminate inter- 
nally, and along which they can run. A theoretical expedient 
is to carry the electrification forward at the proper speed. But 
we want the process to be automatic, so to speak, hence convec- 
tion will not do. Again, if we make the displacement start from 
one portion gf an electrically conducting tube and terminate 
upon the rest, we must insulate the two portions from one 
another, and then there will be a division of the charges 
between the interior and exterior, so that the result will be an 
external as well as an internal wave, or rather, one wave occu- 
pying both regions. 

It would appear that the only way of completely solving the 
problem of the automatic transmission of plane waves within a 
single tube is a theoretical one, employing magnetic as well as 
electric conductance. To see this, imagine any kind of purely 
plane wave being transmitted in the normal manner through 
the ether, and fix attention upon a tube of the flux of energy, 
or a beam, using optical language. This beam cuts perpen- 
dicularly through the reference planes, in which the lines of 
electric and magnetic force lie, which again cross one another 
perpendicularly. The shape of the section of the beam by a 
reference plane may be arbitrary. But let it be quadrilateral, and 
so that it is bounded by magnetic lines on two opposite sides, 
and by electric lines on the other two. Now let the two sides 
of the tube upon which the displacement terminates perpen- 
dicularly, be electrically conductive thin sheets, and the other 
two sides, upon which the induction terminates perpendicularly, 
be magnetically conductive sheets. If the conduction be perfect, 


we . shall not interfere with the transmission of the beam 
within the tube. That is, we have solved the problem stated. 
We may also notice that the external portion of the wave is 
not intrvfered with by the tube. But the interposition of the 
tube in the manner described renders the external and internal 
waves quite independent of one another, and either of them 
may be suppressed. 

Similarly, if the interior region be made finitely conductive, 
electrically or magnetically, or both together, we can still in- 
vestigate the transmission of waves along it in the same way 
as previously described for complete plane waves in a homo- 
geneous medium made conductive, and the same applies to the 
external region, independently of the internal. Going further, 
we may do away with the diffused conductances, and concen- 
trate equivalent resistances in the plates bounding the tubes, 
in the same way as we replace magnetic conductance of the 
external medium by equivalent electric resistance of the wires 
in passing from the exact plane-wave theory to the practical 
theory of wires in terms of V and C. That is, the two plates 
on which the displacement ends may be made electrically resis- 
tive, the resistance taking the place of the magnetic conduct- 
ance in the interior ; whilst the other two plates should be 
made magnetically resistive, if the interior electric conductance 
is also to be abolished. We may then express the propagation 
of waves in the tube in a manner resembling the practical 
theory of wires, though it will no longer be an exact theory. 
Nor will the interior and exterior regions be quite independent 
of one another now that the tube is only finitely conductive. 

Interpretation of Intermediate or Terminal Conditions in 
the Exact Theory. 

207. Leaving these somewhat abstruse considerations, re- 
turn to the practical theory concerning wires and its connection 
with the exact theory involving magnetic conductance. In 
the working out of the practical theory (which is not yet, how- 
ever, the theory of official representatives of practice, though 
they are decidedly getting on), we have often to consider the 
effects due to intermediate insertions of resistance in the circuit 
of the leads, or of shunts across them, and other modifications. 



The question now is, what do these insertions represent in the 
exact theory ? 

Consider first a pair of parallel leads of no resistance. This 
will admit of the application of the exact theory, provided there 
be either no leakage or the external medium be uniformly con- 
ductive electrically. Now we know that when the leads have 
resistance, we make the transformation to the exact theory by 
substituting uniform magnetic conductivity of the external 
medium of the proper amount. Plainly, then, if we insert an 
electrical resistance in a lump in the circuit of the leads at any 
place, we should, in the exact theory, transfer it equivalently 
(changed to magnetic conductance) to the whole of the corre- 
sponding reference plane. That is to say, we must make the 
reference plane (outside the leads) uniformly conductive mag- 
netically, so that the total magnetic conductance of the equi- 
valent plate inserted equals the electric resistance which it 
replaces. We may then investigate the action of the plate on 
the waves traversing it in the manner described in a previous 
paragraph ( 193 and after). 

Similarly, the effect of an electrically conducting bridge 
across the circuit is equivalent to that of an electrically con- 
ducting plate at the reference plane. Their conductances are 
here not only equal, but of the same kind. We merely re- 
arrange the leakage conductance so that it shall act uniformly, 
to suit plane waves. 

Thus, a terminal short-circuit should be replaced by a 
terminal perfectly conductive plate, reflecting H positively and 
E negatively, or with reversal of sign ; or, which is the same, 
C positively and V negatively. This maintains E and V per- 
manently zero at the short-circuit, unless there be impressed 
force there. And a terminal disconnection may be represented, 
in the exact theory, by a terminal perfectly magnetically con- 
ductive plate, reflecting E positively and H negatively, or V 
positively and C negatively, thus maintaining C permanently 
zero at the disconnection. 

A somewhat different kind of transformation is required in 
connection with impressed forces. Suppose, for instance, we 
insert an impressed voltage in the circuit of the leads at a given 
reference plane. How is it to be equivalently transferred to 
the whole plane, or to that part of it outside the leads, so that 


it may generate plane waves of the precise type belonging to 
the given leads ? This is by no means so easy to follow up as 
the previous plate substitutions ; but the following may suffi- 
ciently describe the essence of the transformation. First, when 
the generation of disturbances is concerned, it is not impressed 
electric force e, but its curl, that is effective. We must there- 
fore find the curl of the given impressed force. If the latter 
is half in each wire, acting opposite ways, and uniform across 
their sections by the reference plane, then the curl of e, 
say f, is situated on the boundaries of the wires, and encloses 
them. Having ascertained the integral amount of f, it must 
be transferred to the whole reference plane outside the wires. 
Then the question arises how to distribute it properly. To 
answer this, it may be mentioned that when a distribution of 
f starts into action, the first effect is to generate induction 
following the lines of f. This will be made clear later. If, 
therefore, we distribute the lines of f over the reference plane 
in such a way as to exactly copy the natural distribution of 
the magnetic lines in plane waves having the given leads for 
cores, we shall obtain what we want, viz., the sources so 
distributed as to generate plane waves of the required type on 
the spot. 

By means of the above and similar devices, most telephonic 
and telegraphic problems may be converted to problems in an 
exact plane-wave theory. But we must always be careful to 
distinguish between a theory and the application thereof. The 
advantage of a precise theory is its definiteness. If it be dyna- 
mically sound, we may elaborate it as far as we please, and be 
always in contact with a possible state of things. But in 
making applications it is another matter. It requires the 
exercise of judgment and knowledge of things as they are, to 
be able to decide whether this or that influence is negligible or 

The Spreading of Charge and Current in a long Circuit, and 
their Attenuation. 

208. The detailed application of the principles already 
discussed to the data which occur in practice, or which may 
occur in later practice, would lead us into technical complica- 
tions which would be out of place in the present stage of 

DD 2 


development of this work. Only a few general considerations 
can be taken up here in connection with the theory of plane 
waves. We have seen that there are four distinct quantities 
which fundamentally control the propagation of " signals " or 
disturbances along a circuit, symbolised by R, K, L, and S, 
the resistance, external conductance, inductance, and per- 
mittance ; whilst there are two distinct variables, the transverse 
voltage V and the circuital gaussage C, or the difference of 
potential (in an extended sense) of the leads in a reference 
plane, and the current in the leads. The words voltage and 
gaussage were proposed in 27 to represent electromotive force 
and magnetomotive force, and have been employed experi- 
mentally to see how they would work. They work very well 
for theoretical purposes, but I observe that they are not 
generally or universally approved of, perhaps on account of 
the termination " age," or because of the commencements volt 
and gauss being unitary names. It is open, of course, to any- 
one to find names which shall be more suitable or meet with 
general approval. 

But I stick to inductance. Of all the words which I have 
proposed, that one seems to me to be, more than any other 
except impedance and conductance, the right word in the right 
place, and I am bound to think it possible that some of those 
who prefer the old " coefficient of self-induction," do not fully 
appreciate the vital significance of inductance in a theory of 
intermediate action of a medium. The idea of the direct 
mutual action of current-elements upon one another is played 
out. And as regards the name of the practical unit of induct- 
ance, I think the best name is mac, in honour of the man who 
knew something about self-induction, of course. Many other 
names have been proposed, but none so good as mac. Some 
critic has made etymological objections. But what has 
etymology got to do with it 1 The proper place for etymology 
is the grammar book. I always hated grammar. The teach- 
ing of grammar to children is a barbarous practice, and should 
be abolished. They should be taught to speak correctly by 
example, not by unutterably dull and stupid and inefficient 
rules. The science of grammar should come last, as a study 
for learned men who are inclined to verbal finnicking. Our 
savage forefathers knew no grammar. But they made far 


better words than the learned grammarians. Nothing is more 
admirable than the simplicity of the old style of short words, 
as in the A sad lad, A bad dog, of the spelling book. If you 
transform these to A lugubrious juvenile, A vicious canine, 
where is the improvement ? 

Now, as we have already explained the characteristics of dis- 
tortion of plane waves by conductance in the medium generally, 
of two kinds, acting differently on the electric and magnetic 
fluxes, the corresponding relations in a long circuit of leads 
need be only briefly mentioned. Consider an infinitely long 
circuit of parallel and equal resisting leads under uniform 
external conditions, and governable by the relations laid down. 
Let it be, at a given moment, wholly free from charge or cur- 
rent except at one place, where there is merely a charge, say 
Q . Or, say Q = SV on unit length only at the place. Left 
to itself, this will spread. But it will do so in widely diverse 
manners under different circumstances, although there are some 
common characteristics. The speed of propagation of disturb- 
ances is v = (LS)~*, independent of the values of R and K, the 
dissipation constants. Thus, at time t after the spreading 
begins, we are bound to find the charge, or what is left of it, 
within the region of length 2vt, being vt on each side of the 
origin. Beyond this region there is no disturbance. 

Next as regards the amount of the charge. It cannot in- 
crease, but it may diminish. If the insulation be perfect, which 
means that K = 0, the charge remains constant. It is unaffected 
by the resistance of the leads, although energy is wasted in 
them. It only decreases by the external conductance, or by 
equivalent leakage. Then we have 

...... (1) 

to express the charge Q at time t, decreasing according to the 
well-known exponential law. 

As regards the manner of spreading, either with or without 
the subsidence, this is represented in the exact theory by the 
spreading and subsidence of displacement in a medium which 
is magnetically as well as electrically conductive, although the 
former property does not affect the total amount of displace- 
ment. We see, therefore, that the distortionless case, which 
occurs when the time constants L/R and S/K are equal, forms 


a natural division between two distinct kinds of resultant 
distortion. In this neutral case, the charge Q splits into two 
halves which at once separate from one another at relative 
speed 2v. At time t later they are each attenuated to 

KHiQo^" 8 (2) 

The energy was wholly electric to begin with, but in each of 
the resulting waves it is half electric and half magnetic, be- 
cause they are pure waves. As they attenuate, their energy is 
wasted, and this occurs so that half is wasted by the resistance 
of the leads, and half by reason of the external conductance. 
This is necessary in order that a pure wave shall remain pure. 
The final result is attenuation of the two charges to zero, when 
they are infinitely widely separated, and with this a complete 
simultaneous disappearance of the other characteristics, as the 
magnetic force. In the exceptional case of no resistance and 
no leakage, there is, of course, an everlasting persistence of the 
two charges and of the waves of which they form a feature. 
They go out of range, but not out of existence. In all other 
cases but the distortionless one, the charge (or the remains of 
the original charge) is only partly in the plane waves at the 
two ends of the disturbed region, the rest being diffused 
between them. But this may happen in two ways. The total 
charge at time t is expressed by (1) above. But the charge in 
the terminal waves, half in each, is given by 

Q . -(B/2L+K/28X f ..... (3) 

reducing to (2) doubled in the distortionless case. Therefore 
the excess of (1) over (3) is the amount of the diffused charge. 

It is 

Q .e- K '/ s (l-e-(R/2i--K/2S)*) ) ... (4) 

and is positive or negative according as R/L is greater or less 
than K/S. 

That is, when the resistance of the leads is in excess, the 
charge is positive between the waves as well as in them. But 
when the leakage is in excess the intermediate charge is nega- 
tive. This happens equivalently when the inductance is in 
excess, or the permittance in deficit. How this reversal comes 
about has been already explained in the exact theory. In the 
present circumstances we may say that an intermediate resist- 


ance in the circuit of the leads reflects V positively and C 
negatively, whilst an intermediate leak reflects C positively and 
V negatively. The same applies to the elements of the circuit, 
and from this follows the reversal of V when leakage is in 
excess. Whether the tailing be of the usual positive or of 
the negative kind (referred to the charge), the maximum or 
minimum density is in the middle ; that is, at the origin. The 
density decreases symmetrically both ways to the two ends, 
where the plane waves are to be found. When they have at- 
tenuated to practically nothing, there is left merely the widely 
diffused charge, unless it has attenuated similarly. With the 
resistance largely in excess, there may be nearly all the charge 
left when the terminal waves have become of insensible sig- 
nificance. But if the leakage be in excess, the disappearance 
of the terminal waves is accompanied by a similar disappear- 
ance of the intermediate negative charge, because the total 
charge is always positive. 

So far regarding the spreading of a charge, next consider the 
spreading of induction. The magnetic analogue of Q is P = LC, 
the magnetic momentum per unit length of circuit. So, as L 
is constant, the spreading of P is represented by that of C. 
Suppose initially there be current C in unit length at the 
origin, with momentum P = LC , and no charge there or else- 
where. This also splits immediately into two plane waves, 
which only differ from those arising from a charge in having 
similar C's and opposite V's instead of similar V's and opposite 

The total induction remains constant only if the leads 
have no resistance. In general it decreases, so that at time t 
we have 

P = P .-**0- (5) 

At the same time the amount in the terminal waves, half in 
each, is 

P . c -(*/2L+K/2S)e ...... (6) 

The amount between the terminal waves is, therefore, the 
excess of (5) over (6), which is 

p o -R/L(l_ e -(K/2S-R/2LX) ... (7) 

These equations (5), (6), (7), may be instructively compared 
with the set (1), (3), (4), their analogues. Notice that in 


transforming from one set to the other, we exchange V and C, 
Q and P, R and K, L and S. In the exact theory the R in 
these formulae would also be a conductance, as before described. 

By (7) we see that in the distortionless case all the induction 
left is in the terminal waves, which is to be expected. But we 
require the leakage to be in excess for the intermediate induc- 
tion to be positive, and conversely the resistance must be in 
excess for it to be negative. Of course, the initial induction is 
assumed to be positive. Also observe that leakage by itself 
has no effect on the total induction except to redistribute it, 
just as the resistance of the leads has no attenuating effect 
on the total charge. As before, the maximum or minimum 
density of induction is at the origin, though, of course, the 
terminal inductions may have greater or smaller density than 
this, according to the extent of attenuation at the moment 
concerned. Remember that the destruction of induction by 
the resistance of the leads is represented, in the exact theory, 
by its destruction by the equivalent magnetic conductance 

Putting together the preceding results, we may readily see 
the effect of having both V and C initially at any spot. One 
case is specially noteworthy. Take V = L^C^ initially. This 
means a pure plane wave sheet, travelling in the positive 
direction. There is, therefore, no initial splitting, and the 
wave just goes on. To what extent this will continue depends 
upon the relation the constants of the circuit bear to the 
distortionless state. When the latter obtains, the wave is 
transmitted without spreading out behind, so that at time t 
the initial state, if existent over unit length, will be found 
over unit length at distance vt to the right, attenuated to 

V -K</L, ..... (8) 

where for t may be substituted its equivalent x/v. In all other 
cases there is reflection in transit, and the reflected portions 
travel back, besides getting mixed together, thus making a tail 
of length Zvt, half on each side of the origin, but now, of 
course, without a head at the negative end to balance the one 
at the positive end. 

If the resistance be in excess, the charge in the tail is positive 
and the current is negative, at least initially ; whilst if the 


leakage be in excess, the current is positive and the charge 
initially negative. See 195 to 199 for more details, which may 
be readily translated to suit the present case. The total charge 
subsides according to equation (1) above, and the total induc- 
tion according to (5). Similarly (3) represents the charge in 
the one terminal wave (instead of both, as there), and (6) 
shows the induction in the terminal wave. 

If initially V = - L^CQ, this means a negative wave (going 
to the left) to start with. This needs no separate notice in 

The Distortionless Circuit. No limiting Distance set by it 
when the Attenuation is ignored. 

209. The distortionless state forms a simple and natural 
boundary between two diverse kinds of propagation of a com- 
plicated nature, in each of which there is continuous distortion 
which is ultimately unlimited. Given time enough, and a 
circuit of infinite length to work with, the least departure 
from the distortionless condition would be sufficient to allow 
all the natural changes to be gone through, with ulti- 
mate unlimited distortion. The attenuation is a separate 
matter in this connection, though itself of great importance. 
If, however, we are only concerned with the distortion produced 
in a finite interval of time, which may be quite small, then the 
matter may be differently regarded. There is much or little 
distortion, generally speaking, according as the ratios R/L and 
K./S are nearly equal or widely different. But this needs to be 
understood with caution, for obviously there may be next to no 
distortion of signals even when the distortionless state is widely 
departed from, provided the changes of charge and current are 
made slowly enough. But when this is not the case, and there 
is marked distortion of waves in transit, then we shall increase 
it by making the time-constants more unequal than they are, 
and decrease it by a tendency to equalisation. Furthermore, 
this process is a continuous one, so that, starting from the dis- 
tortionless state connected with equal time-constants, we may, 
by continuously increasing one time-constant, bring on con- 
tinuously increasing distortion of one kind, or else, by increasing 
the other, whilst the first is kept constant, bring on con- 
tinuously increasing distortion of the other kind. And this is 


true by variation of any one of the four quantities concerned, 
the resistance, leakage, inductance, and permittance, to prove 
which we need only remark that with any values given to these 
constants we can equalise the time-constants and abolish the 
distortion altogether by altering any one only of the four. 

When a circuit has been brought to this state, then arbitrary 
signals of any size and manner of variation, originated at the 
beginning of the circuit, will run along it at the speed of light, 
and in doing so suffer no alteration save a weakening according 
to the exponential law given above. The voltage and gaussage 
at any spot will be always in the same phase, and the suc- 
cession of values will faithfully repeat those at the origin. If 
the circuit be infinitely long, the train of disturbances will run 
out to infinity, of course, though it may be still a long way 
from infinity when the attenuation is so great as to make the 
disturbances insensible. Therefore, disregarding the attenua- 
tion, there is no limiting distance of possible signalling. More 
than that, it is not only possible, but perfect signalling, accord- 
ing to this theory. 

But there is no perfection in this world. Even if a circuit 
were constructed with constant R, K, L, and S, and with R/L = 
K/S exactly, so that it should be truly distortionless in this 
practical theory, it would not be so in reality. There are 
several disturbing causes, which, though they might not be of 
much importance in general, would -serve to prevent the attain- 
ment of the clean-cut perfection of the theory which ignores 
them. Instead of zero distortion, then, it is practically only a 
state of minimum distortion that is attainable. But it may be 
a very good imitation of the theoretical ideal. It is unnecessary 
to enlarge upon the necessary failure of a professedly approxi- 
mate theory when pushed to extremes. It is sufficient to say 
that the ignored disturbing influences alone would serve to set 
a practical limit, even if the condition R/L = K/S were truly 
attained, and attenuation were of no importance. 

But there is an influence in full action in the practical theory 
itself which must not be overlooked, namely, the attenuation 
due to resistance and leakage. Of what use would it be to 
have a distortionless circuit if it took nearly all the life out of 
the current in the first 100 miles, when you want it to go 
1,000? Now, on paper nothing is easier than to increase the 


battery power to any extent you want, till, in fact, the minute 
fraction which got to the desired distance became magnified 
up to a recognisable size. But practically there would usually 
be strong reasons against this process. At any rate, it is im- 
possible to overlook the attenuation. It is not merely enough 
that signals should arrive without being distorted too much ; 
but they must also be big enough to be useful. If an Atlantic 
cable of the present type were made distortionless by the 
addition of leakage, there would be no sign of any signals at 
the distant end with any reasonable battery power. Nor can 
we say that telephony is possible on a circuit of given type to 
this or that distance merely because a certain calculable and 
small amount of distortion occurs at that distance. Nor can 
we fix any limiting distance by consideration of distortion 
alone. And even if we could magnify very weak currents, say 
a thousandfold, at the receiving end, we should simultaneously 
magnify the foreign interferences. In a normal state of things 
interferences should be only a small fraction of the principal or 
working current. But if the latter be too much attenuated, 
the interferences become relatively important, and a source of 
very serious distortion. We are, therefore, led to examine the 
influence of the different circuit constants on the attenuation, 
as compared with their influence on the distortion. 

The two Extreme Kinds of Diffusion in one Theory. 

210. Although the equations connecting the voltage V and 
gaussage 0, through the line constants R, S, K, and L, are of 
a symmetrical character, so that, abstractedly considered, a 
study of the nature of their propagation on either side of the 
distortionless state is as desirable as on the other side ; yet 
when we examine the conditions prevailing in practice, we see 
at once that one side only presents itself actively. The reason 
is easily to be seen. It is in all cases desired to get plenty of 
received current, because that is the working agent, and leak- 
age is detrimental to this consummation. There are, besides, 
practical inconveniences connected with leakage. Thus it comes 
about that we are mainly concerned with the state of things 
existing when K/L is greater than, or at the least equal to, K/S, 
and scarcely at all with the exceptional case of an excess of 


leakage, unless it be as a curiosity, and for the sake of its 
theoretical connection with other electromagnetic problems. 

Now this concerns the propagation of V and C through the 
external dielectric. The influence of R prevails over that of 
K, and controls matters. It is, however, somewhat remarkable 
that the other side of the propagation problem does actually 
present itself actively, when we regard the important secondary 
effect of the propagation into the guides of the cylindrical 
waves, which result from the passage over them of the plane 
waves in the external dielectric. Suppose, for example, there 
is no external conductance. This gives one extreme form of 
propagation of V and C, the quantity K/S (or equivalently k/c) 
being zero, and R/L finite. Now, remember that in the exact 
plane-wave theory R stands for the magnetic conductance of 
the external medium, so that R/L is the same as g//j,, and there- 
fore we are, when considering V and C, also discussing the 
influence upon the plane waves of the fictitious property of 
magnetic conductance. 

On the other hand, in the guides themselves, which are elec- 
trical conductors, and perhaps, and most probably, dielectrics 
as well, it is k/c that is greater than g/fj., since the latter is 
zero. Here, then, we have the other extreme form of the 
theory. In the external dielectric k/c is zero and g/jj. (equiva- 
lent to R/L) finite, whereas in the guides g/p is zero and k/c 
finite. The distortion of the external plane waves by the 
resistance of the guides is, therefore, of the opposite kind to 
that of the cylindrical waves in the guides, for E and H change 
places when the form of the distortion is discussed. 

Now, in the guides, on account of their being very good con- 
ductors, the permittivity is swamped, and may be ignored 
in calculating resultant effects. Then we have the theory of 
pure diffusion (as of heat according to Fourier), controlled by 
the two constants k and //,, since both g and c are now zero. 
Similarly, in the external dielectric, it may happen that the in- 
fluence of fj. (or equivalently of L) is insensible, as in the slow 
working of long cables ; then the propagation of V and C is 
controlled by the two constants g and c (or actually and equiva- 
lently by R and S), whilst the other two, k and \^ are zero. 
Here, then, we have a curious exchange of active properties 
between the dielectric and the conductor. In both, the resul- 


tant propagation follows the diffusion law, but is controlled by 
different properties. The propagation of V or E outside is like 
that of H inside, and the propagation of C or H outside is like 
that of E inside. 

We see by the above illustrations that in spite of the absence 
of magnetic conductivity in reality, we are, nevertheless, obliged 
to consider both sides of the wave-theory it involves, even in 
one and the same actual problem, provided it be treated com- 

The Effect of varying the Four Line-Constants as regards 
Distortion and Attenuation. 

211. As, however, we are not immediately concerned with 
the internal state of the guides, and, so to speak, eliminate it 
by treating R and L as constants, we may now dismiss its 
consideration, and return to V and C only, on our understand- 
ing that the quantity K/S is always less than R/L, or does not 
exceed it in the limit. 

Now we have shown that the distortion in transit depends 
upon the difference 

R K. /n \ 

2L-2S (9) 

whilst the attenuation of the front of an advancing wave 
depends upon the sum 

o = A + JL. (10) 

2L 2S 

Both a- and p should be as small as possible, of course. The 
four line-constants act upon p and o- in different ways, and it is 
rather important to understand them. So, remembering that 
the first term of the right members of (9) and (10) is bigger 
than the second, observe the effect of varying the line-con- 
stants, one at a time. 

[R]. By increasing R we increase both p and <r. Thus the 
resistance of the guides is wholly prejudicial, since it not only 
weakens signals in transit, but distorts them. It is, therefore, 
a fundamental notion that the resistance of the guides should 
be reduced if possible, but never increased, if the object be to 
facilitate signalling. 


[L]. By increasing L we decrease both p and or. Thus the 
inductance of the circuit (other things being equal) is wholly 
beneficial, since it diminishes both the attenuation and the dis- 
tortion in transit, or makes the received signals both larger and 
plainer. These are not necessarily insignificant effects, but 
may be very large when applied to very rapid vibrations on a 
long circuit. 

[K]. Increasing K increases p and reduces or. Thus leakage 
is both beneficial and prejudicial, according to circumstances, 
since although it reduces the distortion, it simultaneously in- 
creases the attenuation in transit. This action is most pro- 
nounced when the signalling is slow. 

[S]. Increasing S increases cr and reduces p. It may be, 
therefore, both prejudicial and beneficial. But practically the 
important effect of S, notably on cables, is the prejudicial one 
of distorting the signals, which is a large effect.* 

Now the above statements, though obviously true enough as 
deductions from (9) and (10), are merely qualitative. The 
practical import of varying one quantity when three others are 
constant, depends upon the actual immediate values of all four. 
The length of the line also comes in as an important factor, 
and likewise the frequency of the waves, to settle under what 
circumstances this or that variation of the line constants is 
important in a special case. Nevertheless, the above properties 
are important, and should be most usefully borne in mind in 
general reasoning, as well as in the detailed examination of 
formulae, for instance, the important solutions for simply 
periodic waves. 

Some of these effects have been known and understood from 
the earliest times, or, more correctly, since the time when 
William Thomson taught telegraph-cable engineers the prin- 
ciples of their business (in its electrical aspects), and, to a 
great extent, the practice too. They were previously in quite a 
benighted state, generally speaking, though there were a few, 
whose names it is needless to mention, who combined enough 

* Every rule is said to have an exception. It is, at any rate, possible 
for the above rules to fail, or appear to fail, when the making and reception 
of the signals is not of a simple kind. We may then attribute the failure 
to instrumental peculiarities. An example will occur a little later in which 
K reduces working speed largely. 


electrical science with their practical knowledge to have fairly 
correct ideas about retardation in submarine cables. I do 
not mention Faraday in this connection, for that great genius 
had all sorts of original notions, wrong as well as right, and 
not being a mathematician, could not effectually discriminate, 
especially as he had so little practical experience with cables. 

A complete history of the subject of the rise of Atlantic 
telegraphy, including the scientific side, written by a man fully 
and accurately acquainted with the facts, preferably with per- 
sonal knowledge, and devoid of personal bias, would be of great 
value and of permanent utility. It will not be very long before 
no contemporary will be left to do it. Perhaps, on the other 
hand, it might be more fairly done at second hand. In the 
meantime, there is a somewhat bulky volume, the Report of the 
Submarine Cable Committee of 1859, which serves to show the 
state of knowledge at that date. But it is the history of the 
march of knowledge in the several preceding years that is 

The two effects referred to are that resistance and permit- 
tance (R and S) act conjointly in producing "retardation," so 
that both R and S should be reduced to improve the electrical 
efficiency of a cable. This is embodied in Lord Kelvin's theory 
of 1855, and gave rise to a rather neat law of the squares, 
something that practicians could grasp. This law soon became 
matter of common knowledge amongst cable engineers and 
electricians generally. But when once grasped they could not 
let go, and in later times the law has been most grossly abused 
and misapplied. The fact still remains, however, that R and 
S are prejudicial. But allowance has to be made for other 
influences, and it may be very large allowance. 

But, very curiously, there has been an exception to the rule. 
There have not been wanting electricians who, when telephony 
was extended to circuits containing underground work, sup- 
posed that high, not low resistance, was good for telephony. 
This was giving up the law of the squares with a vengeance. 
But there was really no warrant for such a conclusion, either 
theoretical or practical. As this view turned out manifestly 
wrong by practical failure, it was followed by a precipitate re- 
vulsion, the law of the squares being again seized, and grasped 
more firmly than ever, and against all reason. The fact that R 


and S are prejudicial does not make the law of the squares. To 
have that, R and S must be the only controlling electrical factors. 

All the other effects are of later recognition, and were mostly 
discovered, and their laws investigated, and consequences 
worked out by myself. They completely remodel the subject. 
The old views remain right, in their proper application, bub 
have a very limited application in general, as in modern tele- 
phony particularly. My extended theory opened out a wide 
field for possible improvements in signalling over long circuits 
in advance of what could be possible were R and S the only 
controlling factors, and so far as it goes, modern telephony 
already goes far beyond the supposed restrictions of the law of 
the squares. 

We should not only consider the effects of varying the four 
line-constants one at a time, but also of combinations. Thus, 
K may be counteracted by L. The former is beneficial, inas- 
much as it lessens the distortion, but it is prejudicial in the 
attenuation it causes in transit. But if we simultaneously in- 
crease L, we counteract the attenuation caused by K ; more- 
over, we still further lessen the distortion. This is corrobora- 
tive of the preceding remark as to the unreserved benefit due 
to L. We see, therefore, that [K] and [L] above point the way 
to salvation, if that be by means of increasing the distance 
through which signalling can be carried on with a cable of given 
type as regards R and S. But let it not be forgotten in studying 
K and L, that R and S themselves might perhaps (in some 
particular scheme, for example) be reduced as cheaply as K and 
L could be carried out. Be liberal with the copper, for one thing. 
Look out for some practical way of insulating the wires which 
shall not make the permittance be so great as it is at present 
with gutta-percha or india-rubber. Something of this sort has 
been done for telephone underground cables, though I do not 
know the practical merits. But, as regards Atlantic cables, it 
should be something very good and trustworthy, or the reduced 
permittance may be bought too dearly. 

There has been some difficulty in getting people to grasp the 
ideas embodied in [K] and [L], especially the latter, of course, 
on account of its novelty ; whilst [K] was already partially 
recognised by knowing ones, and is capable of very ready ex- 
planation. [L] was the trouble. Now, remembering the state 


of fog that prevailed in the fifties as regards the effects of 
11 and S alone, there is no reason for wonder that more recon- 
dite effects, as of L, unaccompanied by prior experience, should 
not be readily recognised when pointed out, though that was 
no excuse for burking the matter. [K] and [L], especially the 
latter, require to be studied to be understood. But if people 
will not take the trouble to study the matter, how can they 
expect to understand it 1 Now I find that there has been too 
much of an idea that I have merely given some very compli- 
cated formula which no fellow can understand, and have made 
some dogmatic and paradoxical statements about the conse- 
quences. But a reference to my " Electrical Papers " will show 
that I have treated the matter of propagation very fully, in 
the elementary parts as well as in the advanced, and from 
many points of view, and that I have not been above pointing 
the moral by numerical examination of results, whilst at the 
same time I have been particularly careful in elaborating the 
theory of the distortionless circuit, on account of its casting 
so much light upon the subject generally. The notion that 
has been promulgated, that I cannot make myself understood 
by physicists, recoils upon its authors with great force. Letting 
alone the fact that some physicists have understood me, and 
moreover, the fact that various students have understood me 
(as I know by my correspondence), there is the very notable 
fact that the promulgators of the false idea (which must be 
most discouraging to students) have evidently not taken the 
trouble to examine my writings to see what my views really 
are, but have only made acquaintance with fragments thereof. 
I have advertised the distortionless circuit so often that I have 
become ashamed of doing so. Nevertheless, since it is, as I 
have often pointed out, the royal road to an inner under- 
standing of the subject in general, I do not hesitate now to 
add some more remarks in connection therewith, and leading 
up thereto, in attempted elucidation of former work. We 
should first consider [K] alone, or in conjunction with [R] and 
[S], and then [L], and finally both [K] and [L]. 

The Beneficial Effect of Leakage in Submarine Cables. 
212. What leakage does is to increase the attenuation of 
waves in transit, but at the same time to lessen the distortion 



they suffer. Now there is, by elementary knowledge, no 
difficulty in seeing why leakage on the line should diminish the 
strength of the received current, in spite of the fact that the 
strength of the current sent is increased by the leakage. Ohm's 
law applied to a simple circuit with a shunt attached will 
suffice to give the reason so far as one fault is concerned ; and 
the principle is the same for any number of faults, or for dis- 
tributed leakage. 

As regards the lessening of the distortion, that is not quite 
so elementary. It does not present itself for consideration at 
all in common telegraph circuits. When, however, a cable is 
in question, and there is manifest distortion of signals, the 
influence thereupon of leakage becomes an obvious matter of 
inquiry. We may do it on the basis of the electrostatic theory. 
In fact, in his telegraph theory, Lord Kelvin did insert a term 
in the differential equation to express the effect on the poten- 
tial of uniform leakage, though it was merely in passing, and 
I do not think there was any development of the matter in his 
later papers. The obvious attenuating effect of leakage, and 
the manifest evils of faults, caused high insulation to be aimed 
at, and possibly caused the full effect of leakage to be over- 
looked, and its theory to remain uninvestigated. 

Now a cable is a condenser, or leyden, and a very big one 
too, although its permittance is so widespread that the per- 
mittance of a yard or so is very small. The discharge of 
a condenser against resistance takes time, and so does its 
charge, and the greater the resistance the greater the time 
needed. In the extreme, a perfectly insulated charged con- 
denser keeps its charge unchanged, and so does a well-insulated 
cable, if also insulated at the ends. Even when it is earthed 
(or grounded, as the Americans say) at both its ends, only the 
portions close to the end can discharge at once. Away from 
the ends, the discharge has to take place through resistance, 
to a varying extent though in different parts, when we regard 
the long cylindrical condenser as a collection of small con- 
densers side by side in parallel, with all the similar poles joined 
together by resisting wires. This is how it comes about that 
the resistance acts conjointly with the permittance in causing 
" retardation " in cables, so that the electrostatic time-constant 
is proportional to their product, or RS/ 2 . It is obvious, then, 


that we can facilitate the discharge by lessening the resist- 
ance to discharge, that is, by leakage in the cable, either dis- 
tributed or in detached lumps. At the same time the charging 
is similarly facilitated or expedited. 

Now, in making signals from end to end, the operation of 
charging the cable is necessarily precedent to that of receiving 
current at the distant end, which, in fact, represents a part of 
the charge coming out. Therefore, to expedite changes in the 
charge in the cable, not necessarily or usually complete charges 
or discharges, results in expediting changes in the current at 
the distant end. That is, it lessens the distortion at a given 
working speed, and so expedites the signalling. There is, in 
the electrostatic theory, no limit to this process, and by suffi- 
ciently reducing the insulation resistance, or, which is the same, 
increasing the leakance (which is, perhaps, an admissible short 
name for leakage conductance), we might signal through the 
cable as fast as we pleased, were there no other effect produced. 
This other effect is the extra attenuation. Whilst the possible 
speed of signalling can be raised apparently ad lib., there is 
only an infinitesimal current left to do it with. (Remember 
that the influence of L is not now in question.) It is important 
to recognise that it is a consequence or natural extension of the 
old electrostatic theory, without any appeal to self-induction, 
that you can signal as fast as you like through an Atlantic 
cable of the present type (with leakage added, of course) simply 
by throwing away most of the current, and utilising the small 
residuum that manages to survive the ordeal. 

Thus, a proper Atlantic cable, suitable for rapid signalling, 
should not have high insulation resistance, but the very lowest 
possible consistent with getting enough current through to 
work with, and other considerations, mechanical and com- 
mercial, &c. The theory, moreover, is a sufficiently clear one, 
and does not need elaborate mathematics to make it at least 
probably accurate, though it may, of course, be desirable to 
thoroughly verify the consequences mathematically. 

It is now about 23 years since I came to see, substantially 
as above, the theoretical absurdity of a system of signalling 
which deliberately and seemingly of purpose places in the 
way the greatest possibly hindrance to the efficient trans- 
mission of signals, viz., by the use of the highest insulation 



possible, causing them to be regularly strangled, choked, and 
mutilated, to say nothing of their sometimes dying of inanition. 
But this was mere theory. As a matter of fact, the effect 
of leakage presented itself to me at that time in practice 
on real live cables in a surprisingly different manner. The 
opposition between plausible theory and live practice was so 
glaring that it will be instructive to briefly describe it next. 

Short History of Leakage Effects on a Cable Circuit. 

213. First, as regards long land lines, of say 400 miles in 
length (which is a long distance in this small country), leakage 
presented itself as an unmitigated nuisance. The worst case of 
all was during a continuous soaking rain all over the country. 
When this had gone on for a few hours the insulation resistance 
fell so low, by leakage over the dirty wet insulators mostly, 
though assisted by contacts with wet foliage, &c., that many of 
the important circuits were completely invalided, the received 
current (even with increased battery power) being too small for 
practical use with the instruments concerned. This was usually 
the case in the U.K. on any very wet day. But it was also 
the same in effect in the E. and L, although not so many 
lines were invalided, nor did they break down so quickly. 

Incidentally, it may be mentioned here that the so-called 
"earth-currents "(meaning currents in the Line itself), occurring 
during magnetic storms, such currents being a manifestation 
thereof, did not seem to mind the leakage much, perhaps 
because their E.M.F. was not like that of a battery at one 
end of a line, but acted continuously along the circuit, by a 
continuous slow change going on in the earth's magnetic force, 
or, rather, in the magnetic force in the air surrounding the 
earth. Probably the " earth-current " goes the same way in 
the line as in the earth (provided the line is not set at right 
angles to the real earth-current), whereas when a battery is 
used, the earth and line-currents are opposed. But it is not 
easy in wet weather to get a good magnetic storm, so that close 
observations can be made regarding the influence of leakage on 
earth-currents. We may dismiss them, and sum up that, as 
regards land-lines, leakage does not seem to have a single 
redeeming feature. 


The effects of leakage next presented themselves to me on 
cable circuits with land-lines attached, and were exceedingly 
curious. There were two circuits of the kind, but one in par- 
ticular had a varied history as regards leakage. It was a cable 
of about 400 or 450 miles running east and west, having a land 
line of 120 miles at the east end, and one of 20 miles at the 
west end. The apparatus used was the Wheatstone automatic, 
that is, an automatic transmitter and a polarised receiver, 
recording dots and dashes, and the working was done without 
condensers, by utilising the earth direct to complete the circuit. 

The best results were obtained in fine dry weather with a 
hard frost the harder the better. The insulation of the long 
(east) land line was then at its highest, and the signalling could 
be done up to 30 words per minute or a little more from west 
to east, though considerably less the other way, for a peculiar 
reason. But this was quite exceptional, and did not last long. 
The next best was in any very dry weather. But in ordinary 
average fair weather, not rainy or damp, the speed was much 
lower, say 24 or 20 words per minute. The effect of rain was 
always to lower the speed, perhaps down to 15 words per 
minute. A leakage fault on the land line or a contact with 
another wire had a similar effect, only worse, provided it was a 
bad fault, and the speed would go down to 10 words per minute 
or less. Now it is possible that the extra good results obtained 
in a hard dry frost arose partly from the reduced resistance of 
the east land line (the usual iron wire), but there was no doubt 
at all that the main cause of the wide variations of speed was 
leakage on the land lines, especially on the long one, of course. 

Now it is to be noted here that in the former case of long 
land lines the leakage was prejudicial by the attenuation it 
caused, whereby the received currents were made too feeble to 
suit the apparatus. The working speed, though relatively fast, 
was too slow for any notable distortion of signals in transit by 
the condenser action so marked in cables. But in the latter 
case, of a cable with terminal land lines upon which similar 
leakage occurred, the prejudicial effect was not due to the 
attenuation. There was plenty of current in fact, a super- 
abundance but it did not come out at the receiving end 
properly shaped for making dots and dashes, except at a 
greatly lowered speed. It should also be understood that the 


actual speed of working was always pushed up to the greatest 
possible (the press of business being such as to make 25 hours' 
work per day all too short), and that the automatic character 
of the signalling made the fixation of a limiting speed, without 
complete failure of any marks, quite definite at any particular 
time. Practically, however, the working speed was pushed a 
good deal higher than this limit by making regular use of 
mutilated signals. Now it was proved by daily experience that 
there was a remarkable sensitiveness to damp, or leakage of 
any kind, the speed of working, conditioned by the reception 
of good marks, going up and down like the barometer, though 
with a far wider range than that instrument. The east land 
line was so manifestly the offender that it came in for a great 
deal of blame. Yet it was not a bad line by any means, and 
its leakage would have passed unnoticed, had there not been 
such a peculiar lowering effect on the speed connected there- 

But how about leakage in the cable itself, seeing that the 
above results were so out of harmony with what they should 
have been, by general reasoning regarding retardation ? This 
question was soon answered by the automatic development of a 
first-class leakage fault in the cable. Its effect on the speed 
was substantially the same as that of leakage on the land-line. 
It lowered the speed, and since it was always on, the effect was 
permanent. The ordinary fluctuations disappeared in a great 
measure, as of course did the fine weather high speeds. It was 
like continuous heavy wet. As the fault got very bad the 
speed went down to 10 words per minute or less, in spite of 
increased battery power. 

On describing this prejudicial effect of leakage to the elec- 
trician of the cable ship which came to remove the fault (it was 
the late Mr. S. E. Phillips, of Henley's), he assured me I was 
quite wrong. It was impossible for a leak to act like that. So 
said the captain, too. It was known, said they, that a leak 
was an excellent thing. It had been found so on the French 
Atlantic not long before, and in other cases. I fully believed 
this, because it seemed theoretically sound, and I had heard 
similar statements before. But they did not assist me to an 
understanding of the patent fact that leakage could be highly 
prejudicial instead of beneficial. In corroboration of this, when 


the fault was removed, the speed went up again, and the normal 
state of things was restored. 

The next stage in this history was the arrival of that won- 
derful instrument, Thomson's recorder, which had been recently 
brought out. It was the original Atlantic cable pattern, with 
20 big tray cells for the electromagnets, and a batch of smaller 
ones for the mousemill and driving gear. This instrument 
simplified matters greatly, for the current could be considered 
as and seen to be a continuous function of the time, whereas 
with the polarised recorder of Wheatstone (an instrument quite 
unsuitable for cable work) the variations of the current could 
be only roughly guessed. 

Now, a peculiarity of the siphon recorder was that no definite 
limiting speed was fixable. If one person thought 40 words 
per minute fast enough for good marks ; another, more prac- 
tised, would as easily read them at 50 or 60, and even manage 
to make them out at 70 words per minute, though the last was 
undoubtedly rather troublesome. The automatic transmitter 
being still used, the dashes became big humps, and the dots 
little ones, on the recorder slip, and at 70 words per minute, 
although the big humps were plain enough, the little ones were 
mostly wiped out, though it could be seen where they ought to 
have been, so that some reading was possible. The absence of 
a definite limiting speed under given circumstances introduced 
a fresh complication. Nevertheless, a careful study of the 
recorder slips taken at various speeds, but particularly in the 
neighbourhood of the critical speed for the polarised recorder, 
proved that the plain result of general reasoning that leakage 
should be beneficial in general was certainly true, and, more- 
over, furnished a partial explanation of the anomalous behaviour 
of the polarised recorder. 

The siphon (Thomson) recorder, with its shifting zero, was 
sublimely indifferent to the vagaries of the current which were 
so destructive of intelligible marks in a dot and dash instrument, 
so that ordinary leakage passed quite unobserved as a rule. 
Nevertheless, it was plainly to be seen that when there was 
much leakage the result was to improve the distinctness of the 
marks by reducing the range of variation of the current from 
the zero line, and improving the dots, or small humps, relatively 
to the dashes, or big humps, which were reduced in size. 


But, to clinch the matter, a second leakage fault appeared 
in the cable whilst the siphon recorder was on, and rapidly 
developed. Now, it was only at the west end that there was a 
siphon recorder ; at the east end was the polarised recorder. 
The result was that the speed from west to east went down 
greatly as the fault developed, in accordance with prior know- 
ledge. But the other way, from east to west, receiving with 
the siphon recorder, the working was greatly improved. The 
ultimate result was that when the fault got very bad no marks 
at all could be got at the east end, even by key-sending, owing 
to the great attenuation, whilst we could receive first rate on 
the siphon recorder at the west end. On the very last day, 
when communication from west to east had to be done by a 
roundabout course, we were able to receive a long batch of, I 
think, 150 messages at high speed (about 55 or 60 words per 
minute) by using 80 Leclanche cells at the sending end, and 
putting on the electromagnets all the tray batteries available, 
including most of the small ones, to magnify the marks. Now 
this might seem to be merely a testimony to the sensitiveness 
of the instrument. But the significant point was that although 
the marks were very small, in spite of magnification by increased 
battery power at both ends, they were exceedingly clear. They 
were the best ever got at the speed. We were practically 
working past a fault of no resistance, for the cable soon gave 
up the ghost. The tail end of the big batch finally degenerated 
to a straight line, indicative of a very dead earth. 

On the removal of this fault, the received current was largely 
increased of course. But, simultaneously, the distortion of the 
marks on the siphon recorder became huge at the speed 60. 
It needed to be much lowered to give conveniently readable 

Explanation of Anomalous Effects. Artificial Leaks. 

214. Now the above facts gave a decisive and cumulative 
proof that leakage was beneficial even when concentrated at a 
single spot, provided the current-variations were recorded in a 
simple manner. But a complete explanation of the very large 
decrease in the speed possible when the polarised recorder was 
used, turned out to be rather a complicated matter in detail, 
although sufficiently simple in the fundamental principle. 


This effect, so detrimental to an efficient use of the cable, was 
mixed up with another one, viz., a difference in the working 
speed according to direction, of say 25 per cent, (with the same 
instruments), but variable with circumstances. This effect, 
again, seemed to have two causes, one depending upon the 
circuit, and the uncentrical position of the cable in it, indepen- 
dent of the kind of instruments used ; whilst the other was 
special and peculiar to the automatic transmitter and receiver 
when combined with the uncentrical position of the cable. It 
may be readily imagined that the resultant effect on the 
working speed requires some study to be understood. It will 
be sufficient here to mention the instrumental peculiarity in 
its relation to the detrimental effect of leakage. 

The original automatic transmitter of Wheatstone was con- 
structed to make dots only upon the slip of a special receiver ; 
there were two rows of dots, one set being for the dots, and the 
other for the dashes of the Morse code. But when the inventor 
wanted the E. and I. Co. to take it up, they demanded dots 
and dashes, as usual. So the instruments had to be modified. 
Now I have personal reason to know that a few years later the 
inventor was quite at sea as to the reason of the peculiar effects 
observed in working his instruments on composite circuits and 
cables. It is therefore pretty safe to assume that previously, 
when the conversion was made, the manner of the conversion, 
from making double dots to dots and dashes, was dictated 
purely by considerations of mechanical convenience, and 
this conclusion may be made nearly certain by a comparison of 
the original with the modified transmitter. But Mr. Stroh 
knows all about it. At any rate, the peculiar way of making 
the dash and the long space was the cause of all our trouble 
with leakage, which cost the Company an immense sum of 
money, viz., what they might have gained otherwise, for they 
were blocked with work, and might have had much more. 

To make a dot, a short positive current was sent, immediately 
followed by a similar short negative current. To make a dash, 
after sending the first short positive current, there was an 
interval of no current lasting twice as long as the first current, 
followed by a short negative current which terminated the 
dash. Similarly, to make a long space, a short negative current 
came first, then an interval of no current, and finally a short 


positive current. Now in these intervals of no current, the 
sending end of the line was insulated. When sending current, 
on the other hand, the line was to earth, through the battery. 
Two distinct electrical arrangements were therefore operative 
in turn, viz., a cable with both ends earthed, and a cable with 
only one end (the receiving end) to earth. If the former sys- 
tem always prevailed, the manner of transmission of electrical 
changes would follow one law. If the latter system prevailed, 
it would follow another law, and would give much greater 
retardation. In real fact, the two ways were in alternate action 
irregularly. This has to be considered carefully in examining 
the influence of the uncentrical position of a cable in a compo- 
site circuit, and also of leakage. 

Now the practical effect worked out thus. There was a real 
positive and material advantage in making the dash and long 
space of the Morse code by a short initial current instead of a 
long one, as in ordinary working with reversing keys. I esti- 
mated the gain as amounting to from 100 to 125 per cent, 
under favourable circumstances. This implies that the limiting 
speed is really worked up to. But it was vitally necessary 
to secure this advantage, that the insulation should be very 
good. A peculiar balance had to be preserved between the 
positive and the negative currents. The currents were of equal 
duration and of alternate signs, but necessarily not equidistant 
in time. Now, with exceedingly good insulation, the short* 
current sent at the beginning of a dash was sufficiently prolonged 
in reception as to not merely make a good dash firmly, but to 
keep the balance for the next currents. But if you put a fault 
on the line, and let the charge leak out during the dead 
interval of the dash, then the next following negative current, 
terminating the dash, would be too strong, so that an imme- 
diately succeeding positive current (with a negative current after 
it) to make a dot would miss fire altogether. Similar remarks 
apply to the space. If the negative current that initiated the 
space were allowed to die away by the action of a leak, the next 
following positive current would be too strong, and cause an 
immediately following negative current to miss fire. Remember 
that the reception of the signals was with a polarised recorder, 
the zero of which could not be continuously altered to catch the 
missing currents except by a very artful person who knew what 


was coming, and was quick enough to alter the regulation exactly 
at the right moments. It will, by the above, become roughly 
intelligible how a leak, actually quickening signals in a remark- 
able degree, can yet be most prejudicial in a special system of 
signalling requiring the preservation of a sort of balance. If 
the siphon recorder be substituted the necessity of the balance 
disappears ; but the marks would be very hard to read at the 
same speed. To make them readable the speed should be 
at least doubled. The dashes will then become well-formed 

The reader who may desire to pursue the subject of leakage 
effects further may be referred to my "Electrical Papers," 
Vol. I., especially p. 53 for continuous leakage, p. 61 for more 
details relating to the above-described case, and p. 71 for the 
general theory of leaks, which is fully worked out in several 
cases to show the expediting effect. I also there propose arti- 
ficial leaks. The following extract (p. 77) is easy to read, and 
so may be worth quoting : 

" When a natural fault, or local defect in the insulation, is developed in 
a cable, it tends to get worse a phenomenon, it may be observed, nob 
confined to cable-faults. Under th action of the current the fault is 
increased in size and reduced in resistance, and, if not removed in time, 
ends by stopping the communication entirely. Hence the directors and 
officials of submarine cable companies do not look upon faults with favour, 
and a sharp look out is kept by the fault-finders for their detection and 
subsequent removal. But an artificial fault, or connection by means of a 
coil of fine wire between the conductor and sheathing, would not have the 
objectionable features of a natural fault. If properly constructed it would 
be of constant resistance, or only varying with the temperature, would 
contain no electromotive force of polarisation, would not deteriorate, and 
would considerably accelerate the speed of working. The best position for 
a single fault would be the centre of the line, and perhaps ^ 5 of the line's 
resistance would not be too low for the fault." 

The effect of one leak is marked, perhaps 40 per cent. What 
does this represent in sterling ? How many thousands or 
tens of thousands of pounds' worth of copper and g.p., &c. f 
would be equivalent ? If one leak would not quite do it, then 
two would. 

But to have only one or two leaks is to look upon the leak 
as a dangerous nuisance. Although the best position for a 
given amount of leakance in a lump is in the middle of the 


cable, yet to have the given leakance in a lump is the worst 
arrangement as regards its accelerative efficiency. The best 
arrangement is uniformly-spread leakance, as much as possible, 
with, however, a thoroughly useful residual terminal current 
to work with, which current could be varied far more rapidly 
than is possible with a single leak. 

Is it impossible to find an insulator of comparatively low 
resistance which should be suitable in other respects ? Tele- 
graph cable manufacturers would probably say, No. That, 
however, need not be considered conclusive, because, as is well 
known, when an industry or institution is once established, it 
always gets into grooves, and has to be moved out of them by 
external agency, if at all. But should such an insulator be un- 
attainable, then the only alternative is to have artificial leaks, 
as many as possible. The theory is, I think, quite plain, and 
I know that the practice is also, up to a certain point. But 
here be grooves again. For people working in established 
ways have their own proper work to do, and have no time to 
waste upon " fads." So much is this the case that it often 
happens that they have exceedingly little knowledge of how 
things would work out if they departed from the ruts they are 
accustomed to run along. That is to say, the spirit of scientific 
research, which was to some extent present in the industry in 
its early stages, has nearly all evaporated, leaving behind 
regular rules and a hatred of fads. I am no wild enthusiast, 
having been a practician myself, but it has certainly been a 
matter of somewhat mild surprise to me that cable electricians, 
who have had such unexampled opportunities in the last 20 or 
30 years, say, should have done nothing in the matter of leaks. 
Perhaps there may turn out to be objections at present un- 
thought of. The same could be said of anything untried. 
Several cases of resistance to change of established practice, 
where the changes ultimately turned out to be of great benefit, 
have come under my personal observation. The proposed 
changes were (I believe quite honestly) first scouted as fads, 
and the innovator was snubbed ; next, under pressure, they 
were reluctantly tried ; thirdly, adopted as a matter of course, 
and the innovator snubbed again. But this is the nature of 
things, and can be seen everywhere around very notably in 
the political world. 


Self-induction imparts Momentum to Waves, and that 
carries them on. Analogy with a Flexible Cord. 

215. We have now to consider in what way self-induc- 
tion comes in to influence the transmission of waves along 
wires. The first step in this direction was made by Kirchhoff 
as long ago as 1858, but it is only in recent years that self- 
induction has become prominent as an active agent, and its 
real resultant effect worked out and understood, more or 
less. This effect is, on first acquaintance, a somewhat sur- 
prising and paradoxical one. Everyone knows the impeding 
effect of self-induction in apparatus. How, then, in the name 
of Faraday, can it be beneficial in a telegraph or telephone 
circuit, and make signals bigger and clearer ? To answer this, 
it will be convenient to make use of an analogy in the first 

There is, in a great measure, a formal resemblance between 
the problem of a telephone circuit along which electromagnetic 
disturbances are being propagated, and the mechanical problem 
of the transverse motions of a stretched flexible cord. But to 
make the formal resemblance be also a practical resemblance, 
several little things have to be attended to, and reservations 
made. Now there are a great many mechanical problems con- 
cerning wave-motion subject to friction which make better 
analogies for the use of the mathematical inquirer ; but the 
case of a flexible cord is more or less familiar to everyone, and 
is, up to a certain point, easily realisable. 

The commonest knowledge of the transverse motions of a 
stretched cord is concerning its normal vibrations, the funda- 
mental mode and its harmonics. But we do not want them in 
our analogy. For although we may have something similar 
with short waves, and especially Hertzian waves, it would not 
be easy to set up such a state of things on a telegraph circuit, 
which is usually so long as to prevent the assumption of normal 
modes by reflection at the ends. For the normal vibrations of 
a cord are complex affairs, depending not only upon propagation 
of motion along it, but also on the constraints to which it is 
terminally or intermediately subjected. We desire the propa- 
gational effect to show itself distinctly. But if we disturb a 
stationary stretched wire at one end, the pulses produced run 


to and fro too quickly for practical observation. Take, then, 
a long india-rubber cord, suspended from a height, as is some- 
times done by lecturers. The speed of transmission of pulses 
may be made small enough to allow the disturbances to be 
conveniently watched. Then, if the cord hang vertically at 
rest to begin with, and its lower end be momentarily jerked, 
bringing it immediately back to its original position, a hump is 
generated, which runs along to the fixed end. It is not much 
changed in transit, and is reflected back, and re-reflected many 
times. But we only want to consider the passage of a hump 
one way. Properly speaking, then, we should have so long a 
cord that reflection is (by frictional loss) insignificant. Per- 
haps a very long cord laid upon a surface of ice would be more 
suitable for the purpose. But if the " fixed end " be movable 
transversely, and its motion be resisted by a force proportional 
to its velocity, their constant ratio may have such a value 
given to it (depending upon the mass and tension of the cord), 
that the arriving disturbances do not produce any reflected 
waves, but areabsorbed by the terminal resistance. The cord 
then behaves as if it were infinitely long. This is theory, of 
course, but I dare say a sufficiently good practical imitation 
could be got by means of a terminal block sliding upon a wire 
with friction. An imperfect cancellation of the reflection is 
easily got. 

Disregarding the slow change of type and the attenuation, 
so that there is undistorted propagation of a hump, or of a 
succession of humps making a train of waves going one way 
only, this case is analogous to the transmission of waves along 
a uniform circuit of no resistance. The two constants L and 
S in the electrical case are replaced by another two, viz., the 
density (linear, or mass divided by length) and tension of the 
cord. Thus, large permittance, that is, small elastance, means 
small tension, or a slack cord ; and large inductance means a 
massive cord. If I said a " heavy " cord, it might be thought 
that its weight was concerned in the matter in question, but 
it is not, save as a disturbing factor. The transverse displace- 
ment of the cord may be compared with the transverse voltage 
in the electrical circuit. Other comparisons (theoretically 
better) may be made, but this is perhaps the most convenient 
if the motion of humps be actually watched. 


Now to imitate the action of the resistance (assumed to be 
constant) of an electrical circuit, the transverse motion of the 
cord must be frictionally resisted, and the frictionality (or 
coefficient of friction) should be constant. That is, the 
frictional force should be proportional to the transverse 
velocity. The motion of the cord is frictionally resisted, both 
internally and externally. It is very unlikely that this should 
give rise to a constant frictionality, but there is a similarity 
insofar as the waves sent along the cord do attenuate and are 
distorted as they progress. 

But the friction is not anything like enough in the case of a 
cord in air to imitate the submarine cable ; besides that, the 
mass is too great, and likewise the tension. Even to copy 
telephonic waves going along very long copper wires of low 
resistance, we should increase the friction in the mechanical 
analogue if we desire to have an equal amount of attenuation 
in a wave length. The cord should, therefore, be in a viscous 
medium, and the viscous resistance to the cord's motion should 
be so great as to produce a marked attenuation during the 
transit of a hump, though it should not be so great as to 
prevent its distinct propagation as a hump, without great dis- 
tortion of shape and spreading out. Under these circumstances 
we may exemplify low-resistance, long-distance, overland tele- 
phony, where the inductance is relatively large and the 
permittance relatively small. 

But to imitate a submarine- cable we should have a slack 
cord of small mass in a highly viscous medium. We can then 
send a hump only a little way along it as a hump, for it 
becomes greatly distorted and diffused. Consequently, if we 
compared the displacements at a great distance along the cord 
with those impressed upon it at the free end, we should find a 
state of things quite different from what occurs in the case of 
an india-rubber cord in air. The main features of the originated 
disturbances might be recognised, unless the attenuation were 
too great, but they would be out of all proportion. In particular, 
there is a tendency to obliterate small humps, or ripples whilst 
the big ones persist. The propagation is of the sluggish kind 
characteristic of diffusion, quite unlike that of purely elastic 
waves. If the slack cord could have no mass at all, we should 
then imitate a submarine cable wholly without inductance. 


Under these circumstances we could convert the diffusive 
waves to something like elastic waves by giving mass to the 
cord. Without changing either its tension or the frictionality, 
we could, by a continuous increase of density, cause the nature 
of propagation to pass continuously from the sluggish, diffusive 
kind to the elastic kind with approximate preservation of form 
of waves. We might also do the same by -increasing the 
tension and reducing the frictionality, keeping the density 
small (like, by reducing the permittance and the resistance 
in an electrical circuit, bringing the inductance into impor- 
tance). But these are not immediately in question. By 
increasing the mass, or equivalently the inertia of the cord, 
we impart momentum to the waves, which allows them to 
be carried forward, in spite of the little tension, against the 
resistance of the surrounding medium. 

So it is in the electrical case. By increasing the inductance 
we impart momentum to the waves, and that carries them on. 
This statement is independent of the particular mechanical 
analogy that may be employed for illustration. It is probably 
representative of the actual dynamics of the matter, associating 
L with inertia, and LC with momentum. The soundness of 
this dynamical explanation (made by me some years since in 
asserting and proving the fact concerned) was endorsed by Lord 
Kelvin. And I am proud to add, for the benefit of those who 
may excuse themselves from understanding me by the plea 
that I am a Schopenhauer (!), that Lord Kelvin expressed his 
appreciation of the very practical way in which I had reduced 
my theory to practice. 

Many other dynamical arrangements might be alluded to, 
exemplifying the power of inertial momentum to overcome 
frictional resistance, but it is not easy to get a good one 
exhibiting elastic waves plainly. 

To illustrate the nature, and to emphasise the possible 
magnitude of the effect, suppose we have the power of 
continuously increasing the inductance of an Atlantic cable 
from a small quantity up to a very large one, without altering 
its resistance or permittance, and without introducing any 
other actions than those to be controlled by resistance, 
permittance, and inductance. Starting with L = 0, and making, 
say, 1,000 waves per second \t one end, there will be 1,000 


waves per second of workable magnitude only in a small part of 
the cable near the sending end ; the amplitude will attenuate 
to insignificance, not exactly in the shore-end, but before 100 
miles is traversed. But let waves of the same size (which will 
need greater battery power to produce) be kept up at the 
sending end whilst L is continuously increased all along the 
cable. The region of practical waves will continuously extend 
itself right through the cable, by a partial conversion to elastic 
waves, with little attenuation in transit. With very big 
inductance the waves would reach the distant end nearly in 
full size. Even then they could be doubled in size by the 
reflection, though there would be no useful purpose served by 
that. The amount of inductance required, however, to produce 
these results would be out of all reason. Nevertheless, the 
illustration serves its purpose in showing the action of self- 
induction, which is, as before stated, wholly beneficial, as it 
increases the amplitude and lessens the distortion. Some 
remarks on increasing inductance will come later. 

Self-induction combined with Leaks. The Bridge System 
of Mr. A. W. Heaviside, and suggested Distortionless 

216. In the meantime, some few remarks on self-induction 
combined with leakage should come in here. Self-induction 
came in with the telephone. That was, originally, a mere 
house-to-house affair. At any rate, lines were only a few miles 
long. So condenser action was out of practical question then. 
On the other hand, the frequency of the currents concerned is 
very great, and that makes inertia important. So the theory 
of telephones is a magnetic affair when the lines are short 
enough, just as if they were lines in a laboratory. 

But the telephone soon got beyond that, and it became 
necessary, with the use of longer and longer lines and under- 
ground lines, to examine the influence of their permittance in 
conjunction with their inductance. In hardly any case of 
telephony can we ignore the self-induction, and take the 
permittance alone. The resistance would need to be so large 
that the lines would be wholly unsuitable for telephony, except 
for mere local lines. So permittance and inductance usually 



go together in considering telephony, and if one of them is 
ignorable it is the permittance rather than the inductance. 

Following the telephone came exchanges. But there was an 
exchange before telephones. My brother, Mr. A. W. Heaviside, 
had an intercommunication exchange at Newcastle, using 
alphabetical indicators, whereof the manipulation resembled 
that of the hurdy-gurdy. These were beautiful instruments 
mechanically ; but, of course, the telephone soon stopped their 
manufacture. The exchange referred to was then turned into 
a telephone exchange. Now, in certain cases it was needed to 
have many instruments in one circuit, with local intercommu- 
nication ; a sort of family arrangement, so to speak ; for 
example, a number of coal pits and colliery offices belonging 
to one firm. The custom was to set all the intermediates in 
sequence (otherwise, but perhaps not so well, called in series) 
In the circuit. Why was this done? Simply because they 
knew no better. It was the custom to do so in the telegraphs 
generally, being an obvious arrangement not so much in 
England, though, as elsewhere. But when this was done with 
telephones, or with call-instruments replacing them, it was 
found that the intermediates had a very deleterious influence, 
which very soon set a limit to the admissible number of inter- 
mediates on a circuit and to the length of the circuit, especially 
when underground wires were included. My brother, who took 
up the telephone with ardour from the first (making the first 
one made in England, I believe), found by experiment that it 
was quite unnecessary to put the instruments in sequence 
according to old practice, and that an immense improvement 
was made by taking them out of the circuit, and putting them 
across it, as shunts or bridges. Like lamps in parallel, in fact. 
It seems very easy and obvious now. But it is not so easy to 
conceive the state of mind of old stagers. Anyhow, the pro- 
position to put the instruments in bridge was pooh-poohed at 
first, not being understood. As, however, it was demonstrably 
a wonderful improvement it was adopted, first at Newcastle, 
and then elsewhere, and became known as the Bridge System. 

When this system was brought to my knowledge, and I was 
asked for theoretical explanation, it presented itself in a double 
aspect. First of all, there was the complete removal from 
the circuit of intermediate impedance in big lumps. This was 


manifestly the main reason of the increase in working distance 
possible, and why a far greater number of intermediates co^ld 
be put on. For they were not put in the circuit, but on it, st> 
to speak. 

The other aspect of the matter was this. Considering that the 
bridge system consists electrically of a circuit with a number of 
leaks on it, and bearing in mind my old investigations concern- 
ing leaks, I inquired whether the intermediates were not ac- 
tually beneficial to through communication, independently of 
the previous effect. Was not the articulation better, not merely 
than if the intermediates were in series in the circuit, but better 
than if the shunts or bridges were removed altogether, leaving 
insulation ? My brother's answer was Yes, certainly, in certain 
cases, b*ut doubtful in others. It appeared, however, to be only 
a minor effect, of not much importance compared with the major 
one. Besides, the bridges caused a weakening of the intensity 
of the speech received when many bridges were passed. To pre- 
vent this weakening becoming inconveniently great, the inter- 
mediate call-instruments in bridge were purposely made to have 
considerable resistance and inductance, far more than would be 
needed or desirable for their natural use. This tended to 
prevent the currents passing along the line from entering 
the shunts, and especially so as regards the currents of high 
frequency, and allowed them to be transmitted in greater 

It will be observed here that the expediting effect of leakage 
considered before ( 213, 214) is merely incidental, and is, in 
fact, partly destroyed by the large impedance of the bridges, 
in order to mitigate the attenuative evil. The shunts are 
inductive shunts, intentionally very much inductive. Now, 
my proposed leaks (214) for submarine cables were non- 
inductive, and intentionally so, though it may be remarked 
that it would make very little difference with the slow signalling 
of Atlantic cables if they were inductive. On the other hand, 
a recent proposal of Prof. Silvanus Thompson is to use induc- 
tive shunts on cables. The arrangement is electrically as in 
the bridge system, but the object is different. This point will 
be returned to. 

The investigation of this bridge system of telephony suggested 
to me the distortionless circuit, as I have before acknowledged 



("Electrical Papers," Vol. II., p. 402). It may be of interest 
to state how it came about, since it is not presented to one by the 
above-described experiences with the bridge system. The ques- 
tion was, What was the effect of a bridge, or of a succession of 
bridges, when the self-induction of the line was not negligible ? 
Now the influence of the bridges can be very readily stated for 
steady currents, of course. There is no difficulty, either, in 
finding the full formulae required to express the voltage and 
current and their variations in any part of the system when 
the inductance and permittance are operative, as well as a set 
of leaks, when one knows how to do it. But the general 
formulae thus obtained are altogether too complicated to be 
readily interpretable. I was, therefore, led to examine the 
effect produced by a leak on a wave passing it of the" elastic 
type. This implies that the resistance of the line is sufficiently 
low for waves of the frequency concerned to retain that type 
approximately. (Ln should be greater than R.) Now this was 
comparatively an easy problem, and the result was to show that 
a leak (or conductance in bridge) had the opposite effect on a 
wave passing it along the line to that of a resistance inserted 
in the circuit at the same place, in this sense : A resistance in 
the circuit reflects the charge positively and the current 
negatively, or increases the charge and reduces the current 
behind the resistance. On the other hand, a leak reflects 
the current positively and the charge negatively, or in- 
creases the current and reduces the charge behind the 
leak. (See also 197, 209.) A resistance and a leak 
together at the same place therefore tend to counteract one 
another in producing a reflected wave, leaving only an at- 
tenuating effect on the transmitted wave. This led to the 
distortionless circuit. For although the compensation of the 
resistance in the circuit and the conductance across it is im- 
perfect when they are finite, it becomes perfect when they are 
infinitely small. They may then be a part of the circuit itself, 
namely, the resistance a part of the resistance of the line, and 
the conductance a part of the leakage. We now have perfect 
compensation with this one, or any number of successive resist- 
ances and leaks of the same kind, provided their ratio is pro- 
perly chosen. This occurs when R and K, the resistance and 
conductance per unit length of circuit, of the line and of the 


leakage respectively, are taken in the same ratio as the two 
other constants, L and S. Or, when L/R = S/K, indicating 
equality of the electric and magnetic time-constants. There is 
now an undistorted transmission of any kind of signals. And 
since every circuit has R, L, and S, it follows that it can be 
made distortionless by the addition of leakage alone of a certain 
amount, or distortionless to the extent permitted by the ap- 
proximate constancy of R, L, and S. (See " Electrical Papers," 
Vol. II., pp. 119 to 168, for detailed theory of the distortionless 

In 213 we saw that even in the electrostatic theory leakage 
would allow us to signal as fast as we liked provided we could 
do it with an infinitesimal current. This was not because the 
circuit was made distortionless, but on account of the infinitely 
rapid charge and discharge. Now the inclusion of the in- 
fluence of self-induction shows that we can do the same with a 
finite leakage conductance, and have finite distortionless sig- 
nals, due to waves of the purely elastic type. Given R and S 
then, if L should be so small that the introduction of K of the 
proper amount to reach the distortionless condition should 
produce unreasonable attenuation, then one way of curing this 
would be to increase L. For then a smaller amount of leakage 
will serve. The bigger L, the smaller need K be, and when it 
is very big, then very little K is needed. This brings us round 
to the case at the end of 215, where we had elastic waves 
with little attenuation on an Atlantic cable got without any 
leakage, and since we see now that very little leakage is needed 
to bring about the distortionless condition, we conclude that it 
is approximately distortionless without the leakage. 

Evidence in Favour of Self-induction. Condition of First-Class 
Telephony. Importance of the Magnetic Reactance. 

217. Let us now ask what is the evidence in favour of the 
beneficial action of self-induction. There was plenty in 1887; 
it is overwhelming now. But there are two sorts of evidence, 
the theoretical and the experiential. The two are not funda- 
mentally different, however ; only the former is more indirect 
than the latter. We have certain electrical laws established, 
mainly by laboratory work followed by mathematical inves- 


tigation, and there is no reason whatever to suppose that they 
do not hold good out of doors as well as inside. The important 
thing is to correctly recognise the prevailing conditions and 
determine their results. Now, if you can be pretty sure that 
you have done this, and have confidence in your investigations, 
then you may consider that you prove a thing, even though 
there may be no direct evidence. Discoveries are frequently 
made in this way. " A foundation of experimental fact there 
must be ; but upon this a great structure of theoretical de- 
duction can be based, all rigidly connected together by pure 
reasoning, and all necessarily as true as the premises, provided 
no mistake is made. To guard against the possibility of mis- 
take and oversight, especially oversight, all conclusions must 
sooner or later be brought to the test of experiment ; and if 
disagreeing therewith, the theory itself must be re-examined, and 
the flaw discovered, or else the theory must be abandoned." 
(Oliver Lodge.) But to convince other people is quite another 
matter. They may not be competent to understand the 
evidence. Or they may be fully competent, but not have suffi- 
cient acquaintance with the subject ; or not have time to 
examine it ; or not have the energy ; or have no interest in it. 
Then, as Elijah said to the priests of Baal, you must " Call him 

Now, the foundation of fact is the experimental facts of elec- 
tricity and the laws deduced therefrom, though these laws are 
sometimes more comprehensive than the facts upon which they 
are based. But there was, in 1887, a certain amount of direct 
evidence as well, quite enough to make the theoretical con- 
clusions practically certain, by the confirmation they afforded. 
Assuming the electrostatic theory to be true in telephony (for 
which, however, there waa never any sufficient warrant) we can 
calculate its consequences. They indicate that the product 
RSJ 2 of the total resistance and the total permittance limits 
the distance I through which telephony is possible. How big 
it may be is questionable, on account of the very complicated 
nature of the problem when the human is taken in as a part 
of the mechanism. We cannot say certainly beforehand how 
much distortion is permissible before the human will fail to 
recognise certain sounds as speech. But the electrostatic 
theory shows such very considerable distortion even when 


RSI 2 is as low as 5,000 or 7,000 ohms x microfarads, that it 
is scarcely credible that any sort of telephony could be prac- 
ticable if it were much bigger, as 10,000 for example. Perhaps 
5,000 might be taken as the practical limit for rough purposes. 
But experience showed that good telephony was got up to 
10,000 and higher, in this country with circuits including 
underground cables, as well as in America, where long-distance 
telephony had already made a good beginning. Manifestly the 
electrostatic theory was erroneous by direct as well as indirect 
evidence. Now the inclusion of the influence of self-induction 
in the theory explained these results (and also the failure of 
iron wires and the success of copper) and pointed out how to 
improve upon them, viz., by directing us to lower R and 
increase L in order to increase the value of RS 2 possible. So 
there was sufficient direct evidence then to satisfy anyone com- 
petent to judge who considered it fairly without being pre- 
judiced by the law of the squares. Since that time, the 
amount of this direct evidence has been greatly multiplied, 
and RS 2 has got up to 50,000, which may be about ten times 
as great as the probable value under purely electrostatic con- 
ditions. But there is nothing critical about 50,000. It might 
be 500,000 if the circuit were long enough. We may need to 
reduce R when we increase Z, but not in proportion to the 
inverse square of Z, or anything like it. 

Any long circuit may be made approximately distortionless 
if the magnetic side of the phenomena can be made important, 
and without the inconvenience of the leakage needed to remove 
the distortion, which leakage, however, is essential when the 
magnetic side cannot be made important. The useful guide in 
considering telephony is the size of the quantity R/Ln, the 
ratio of the resistance to the reactance, the two terms appear- 
ing in (R 2 + L 2 tt 2 )*, the impedance of unit length of circuit, 
apart from the permittance. 

The term " reactance " was lately proposed in France, and 
seems to me to be a practical word. It may be generalised to 
signify the value of Lrc, positive or negative, of any combina- 
tion of coils and condensers, L being then the effective induct- 
ance. When L and the reactance are positive, the magnetic 
side prevails and the current lags ; and when it is negative, 
the electric side prevails and the current leads. 


At present, however, R and L are as before. Now the ratio 
R/L/7& may be very large. If so, the influence of L is small, 
and we shall have something like the electrostatic theory, 
provided the range of n be such as to make the ratio always 
large. But this state of things is quite unsuitable for long- 
distance telephony. The most suitable state is when R/Lw- is 
a proper fraction throughout the whole range of n ; that is, 
when R is less than the smallest value of Im. We shall then 
have approximately distortionless propagation of the type 

as may be seen by examining the solution for simply periodic 
waves, and observing that when R/Lw- is small we have nearly 
the same attenuation in a given distance at all frequencies (not 
too low), so that there is little distortion, and the element of 
frequency tends to disappear altogether, and we reduce the 
formula to the above type. 

Practice shows, however, that this state of things is unneces- 
sarily good for commercial telephony, which admits of a 
considerable amount of distortion. Then R/Lw, is not less 
than 1 throughout the whole range of n, but is greater at the 
low frequencies and less at the higher.* For example, if the 
lowest value of n be 625, and L = 20 (centim. per centim., usual 
units), the least value of IM is 12,500. If also the resistance 
is 5 ohms per kilometre, then R is 50,000. The value of 
R/L is then 4 at the lowest frequency, and 2, 1, J, , J at 
the five successive octaves, the last being at n= 20,000. Here 
we have an excellent state of things from n= 2,500 up to 
20,000, and only at the low frequencies does R prevail over 
Lra. If R be halved, or L be doubled (for it is indifferent 
which is done), then R/Lra is less than 1 from = 1,250 
upwards. Divide n by 2?r to obtain he frequency. This 2?r 
has nothing to do with the ridiculous 4?r which the B.A. 
Committee want to consecrate. The significant point in calcu- 
lations of this kind is the position of n in the scale of frequency 
which produces equality of R and Tun. When it is low we 
have good quality ; when high, then relatively bad. Perhaps 
the American electricians may be able from their extended 

* See my " Electrical Papers " for details ; especially App. C, p. 339, 
Vol. II. 


experience to give practical information on this and other 
points concerned in telephony, and construct an empirical 
formula of a sound character for determining the limiting dis- 
tance roughly in terms of the electrical data, with given kinds 
of instruments for transmission and reception. It should be 
understood that the above remarks do not apply to short lines, 
at least in general. We may, indeed, so arrange matters that 
the above type is still preserved (with R/Lw, less than 1) by 
practical annihilation of reflected waves, but otherwise serious 
modification may be needed. And with very short lines we 
may apply a purely magnetic theory, of course. 

Various Ways, good and bad, of increasing the Inductance 
of Circuits. 

218. Supposing, now, that practicians have come to discard 
the supposed limitations so persistently asserted by official 
electricians in England, and to open their eyes to the part 
played by self-induction, they will naturally recognise the wide 
possibilities that are suggested in future developments, and 
want to know whether such possibilities can be converted into 
actualities, and whether the conversion is practicable. For 
myself, I am not much concerned in this part of the question. 
It is for practicians to find out practical ways of doing things 
that theory proves to be possible, or not to find them if they 
should be impracticable. Nevertheless, since I am responsible 
for these views concerning the parts played by self-induction 
and leakage, I add a few remarks on the applicational side. 

The ideal is the distortionless circuit, requiring both self- 
induction and leakage. If we utilise the present existent 
inductance only, then the needed leakage may be too great by 
far in many cases. On the other hand, if the existent induct- 
ance is big enough to make R/Lw. small, in the sense described 
in 217, we do not want the leakage particularly, or not at all 
practically, it may be. It is clear, then, that we should aim 
at making L/R, the magnetic time- constant, as big as possible. 

Now, here is one way of doing it. If, a few years ago, an old 
practitioner was asked to put up a long-distance telephone circuit, 
he, being under the belief that R should be high, or at least 
that it need not be small, and that L should be as small as 


possible, in order to prevent the deleterious action of self- 
induction, would probably carry out his principles by putting 
up a circuit, several hundred miles long, consisting of a pair of 
fine wires, put quite close together. Or, if he was free from the 
error as regards R, he would use larger wires. But still close 
together, for that makes L the smallest. Also, he would not 
consider this to be bad as regards S, the permittance, being 
under the belief that the permittance of the circuit of two wires 
was just one half that of either with respect to the earth, so 
that their proximity did no harm in increasing condenser action, 
whilst it destroyed the deleterious self-induction. Now, such 
a circuit might do very well for local telephony, but for the 
purpose required there could not be a worse arrangement. 
The supposed case is not an extravagant one, since it involves 
the carrying out of official views thoroughly. The cure would 
be quite easy. Increase the inductance by separating the wires 
as widely as possible or convenient. If necessary, reduce R as 
well. The permittance will also be reduced in about the same 
ratio as the inductance is increased. By this simple process L 
may be largely multiplied, say from 2 to 20 (cm. per cm.), a 
tenfold magnification, without separating the wires so much as 
to require a double line of poles. This way of doing it is the 
secret of the success of the long-distance telephony which arose in 
America when iron was discarded. L is made important, and 
R is kept low. But although L increases very fast on initial 
separation of wires, it does not continue to increase fast on 
further separation, so that practical limitations are soon set to 
the process. The above example shows how a very bad line 
may be turned into a very good one by increasing the inductance. 
Another way is suggested, but only to be immediately 
rejected. It is, nevertheless, instructive to study failures. 
We can increase L to any extent by using finer and finer wires. 
This would be admirable if it were all. But, unfortunately, 
this process alters R, and faster still ; so that R/L increases 
instead of diminishing. Consider, for simplicity, merely a wire 
with a concentric tube for return. Then R/L is nearly propor- 
tional to RS. Now Lord Kelvin gave a formula long ago, 
showing that this quantity was made a minimum when the 
radii of the tube and wire bore a certain ratio. Then L/R is 
a maximum, approximately, and on decreasing the radius of 



the wire it falls off, ultimately very rapidly, which is the 
reverse of what is wanted. 

A third way must also be immediately rejected. By using 
iron wires we can increase L largely with steady currents. 
But then the skin effect, or imperfect penetration, becomes 
important with telephonic currents in iron wires. It does two 
things, both harmful. It lowers L and increases R. So we 
do not get the large increase of L suggested at first, but only 
a comparatively small one with currents of great frequency, 
and we do get a lot of extra resistance that should particularly 
be avoided. Iron wires are failures for long-distance telephony, 
as was found long ago. 

The case of copper-coated iron or steel wires is doubtful. 
Such wires were used in America, and it was said that they 
were successful for long-distance telephony. But here, evidently, 
the copper is very important in giving conductance. As regards 
the interior iron core, it must be considered very questionable 
tvhether it can be so good as an amount of copper of the 
same resistance (steady). It is not well placed for increasing 
the inductance, and the imperfect penetration will operate 

If we put the iron outside the copper the iron will be better 
situated for increasing L for steady currents (which is no good), 
but will act injuriously with rapidly oscillating currents, and 
partly prevent the utilisation of the conductance of the copper, 
if it be thick enough to make the increased L important. 

The above considerations show that if iron be introduced to 
increase the inductance it should not be in the main circuit, 
but external to it. There are, then, two principal ways sug- 
gested. First, to load the dielectric itself with finely-divided 
iron, and plenty of it. This is very attractive from the 
theoretical point of view, as it results in the production of a 
strongly magnetic insulator, which is practically homogeneous 
in bulk, being, therefore, a sort of non-conducting iron of low 
permeability. In this way L, as compared with the same with- 
out the iron, may be multiplied greatly. There is a partially 
counteractive influence, however, due to increased permittance. 
This plan is interesting, in view of the wave theory. It is like 
multiplying the p in ether, and lowering the speed of pro- 
pagation, but at the same time allowing waves to keep up 


against the resistance of wires they may be running along. But 
as regards the practicability of this plan I say nothing, except 
that I have often smiled at this ironic insulator (in more 
than one sense ironic), and the idea of telephoning through a 
core resembling a poker with a copper wire run through the 

The other way is to put the divided iron outside the working 
dielectric. Then it had better not be divided into particles, 
but in the way well known to dynamo and transformer people, 
cut up so as to facilitate the flux of induction, whilst being 
electrically non-conducting transversely thereto. 

Now it may be said that submarine cables of the present type, 
having cores surrounded by a large quantity of iron, have large 
inductance already. But it does not follow. If the iron was in 
a solid tube, then undoubtedly L would be much magnified 
with steady currents. But it is not a solid tube, as it consists 
of spirally laid separate stout iron wires. The magnetic con- 
tinuity is interrupted, and the permeance is much reduced, as 
Prof. Hughes pointed out (equivalently) some years ago. But 
besides that, when the current is not steady, we do not utilise 
the iron even in the above imperfect manner. This is par- 
ticularly the case with telephonic currents, when imperfect 
penetration will still further reduce the effective inductance. 
So the increased L due to the sheath must be far less than 
large quantity of iron might suggest. Again, there is the the 
increased resistance due to the sheath not being properly 
divided to prevent it. The case is therefore a considerably 
mixed one. Whilst we can say with confidence that with the 
very slow signalling which obtains through an Atlantic cable 
the sheath cannot make much difference, it is not easy to 
decide whether its action is beneficial with telephonic currents to 
any great extent, though it would be if there were no increased 
resistance. It is not even easy to say what is the L of a sub- 
marine cable under the circumstances. A least value is readily 
obtained, but the magnitude of the increase due to the iron and 
the outside return current is rather speculative. It may be 
well deserving of attention to change the type of the iron 
sheathing, so as to increase L and keep down R. Remember 
here, that although there is plenty of evidence of the beneficial 
(and very important) influence of this principle when it is 


carried out in a direct and natural manner, there is, so far as I 
am aware, no experimental evidence yet published of the equi- 
valent success of indirect ways involving the use of iron. 

Another indirect way is this. Instead of trying to get large 
uniformly spread inductance, try to get a large average induc- 
tance. Or, combine the two, and have large distributed 
inductance together with inductance in isolated lumps. This 
means the insertion of inductance coils at intervals in the 
main circuit. That is to say, just as the effect of uniform 
leakage may be imitated by leakage concentrated at distinct 
points, so we should try to imitate the inertial effect of uniform 
inductance by concentrating the inductance at distinct points. 
The more points the better, of course. Say m coils in the 
length /, or 2m coils of the same total inductance, and 
therefore each of half the inductance ; or mn coils each of 
one 7i th the inductance of the first. The electrical difficulty 
here is that inductance coils have resistance as well, and if this 
be too great the remedy is worse than the disease. But it would 
seem to be sufficient if the effect of the extra resistance be 
of minor importance compared with the effect of the increased 
inductance. This means using coils of low resistance and the 
largest possible time-constants. For suppose 4 ohms per kilom. 
is the natural resistance, and there be one coil per kilom. 
having a resistance of 1 ohm. This will raise the average 
resistance to 5 ohms per kilom. ; and if the time-constant be 
big enough, the extra inductance may far more than nullify 
the resistance evil. The same reasoning applies to coils at 
greater intervals, only of course in a more imperfect manner. 
To get large inductance with small resistance, or, more generally, 
to make coils having large time-constants, requires the use of 
plenty of copper to get the conductance, and plenty of iron to 
get the inductance, employing a properly closed magnetic circuit 
properly divided to prevent extra resistance and cancellation 
of the increased inductance. This plan does not belong to the 
category of those mentioned before, which a moment's con- 
sideration showed to be worse than useless. It is a straight- 
forward way of increasing L largely without too much increase 
of resistance, and may be worth working out and development. 
But I should add that there is, so far, no direct evidence of 
the beneficial action of inductance brought in in this way. 


The combination with leaks does not need any particular 
mention in detail after what has been already said relating 
to the distortionless circuit, which is the theoretical ideal, 
any approach to which is desirable if it be done without too 
much loss ; bearing in mind, too, that we should increase L 
first in preference to putting on leaks first.* I have confined 
myself so far entirely to my theory of 1886-87 and results 
thereof. Inductive shunts involve some other considerations, 
and a modified theory. 

Effective Resistance and Inductance of a Combination when 
regarded as a Coil, and Effective Conductance and Per- 
mittance when regarded as a Condenser. 

219. In any electromagnetic combination the effects of 
electric and magnetic energy are antagonistic in some respects. 
Or we may say that magnetic inductances tend to neutralise 
electric permittances. Or, more generally, that the effects of 
elastic compliance and of inertia tend to neutralise one another. 
Thus, when a coil is under the action of an impressed simply 
periodic voltage, the current lags, owing to the self-induction. 
But if a condenser be introduced in sequence with the coil, the 
lag is diminished, and may be reversed or converted into a lead. 
The result on the current is the same as if we substituted for 
the coil another of the same resistance and of reduced in- 
ductance, or even of negative inductance. Thus, the resistance 
operator of the coil being R + L^, where R and L are its 
resistance and inductance, and p is the differentiator djdt ; and 
that of the condenser being (Sp)" 1 , where S is its permittance, 
when the two are in sequence the resistance operator of the 
combination is 

Z = R + Lp + (Sp)-i, (1) 

which, in a simply periodic state of frequency n/'2ir t making 
ni, becomes 

R + /L-AV . . 

* But remember the influence of the frequency in conjunction with the 
inductance. Leakage can be very beneficial when self-induction does next 
to nothing, as before described. 


This shows that the effective inductance of the condenser, 
which is - (S/& 2 )" 1 , is additive with real inductance when in 
sequence therewith, and being negative, reduces the effective 
inductance of the combination from L to L - (S^ 2 )" 1 . 

But when the condenser and coil are in parallel, it is more 
convenient to use the conductance operators. Thus 

Y = (R + L^)- 1 + Sp ..... (3) 

is the conductance operator of the coil and condenser in parallel, 
being the sum of the conductance operators of the coil and con- 
denser taken separately. And when p = ni, we convert Y to 



from which we see that the combination is equivalent to a 
condenser of conductance R (R 2 + L 2 ^ 2 )" 1 and of permittance 
S - L (R 2 + L 2 ^ 2 )" 1 . The steady conductance being R" 1 , we see 
that the effective conductance is reduced by the self-induction. 
At the same time the effective permittance is reduced, and may 
become negative. Remember that in (3) and (4) the standard 
of comparison is a condenser, not a coil ; whereas in the former 
case (1), (2) the standard of comparison is a coil, not a con- 
denser. For further information regarding resistance and con- 
ductance operators sec " Electrical Papers," Vol. II., p. 355, and 
elsewhere. The present remarks are merely definitive and 
introductory to the theory of waves sent along a circuit, either 
naturally, or partly controlled by subsidiary arrangements. The 
comparison with a coil is, I believe, more generally useful. 
Then we imagine any combination to be replaced by a simple 
coil whose resistance and inductance are those (effectively) of 
the combination. But there are cases when the other way is 
preferable. Then we imagine the combination replaced by 
a condenser whose conductance and permittance are those 
(effectively) of the combination. And in the following it will be 
convenient to use both ways in the same investigation. 

Inductive Leaks applied to Submarine Gables. 
220. Now imagine the permittance of a submarine cable to 
be concentrated in lumps at a number of points. Let S be the 


permittance of one of the equivalent condensers. If it be 
shunted by a coil of such R and L that S = L (R 2 + lAi 2 )" 1 , we 
know by the above that the state of the cable, when simply 
periodic, will be the same as if the condenser and coil were 
removed and replaced by a simple leak of conductance 
R (R 2 + L 2 ?! 2 )"" 1 . This reasoning applies to every condenser, if 
it have its appropriate coil, or to any selected group we please. 
But if the frequency be changed the permittance will come into 
sight again, of the real or positive kind, or of the fictitious 
negative kind, as the case may be. We have, therefore, the 
power of reducing the effective permittance of a cable under 
simply periodic forces, by means of numerous auxiliary induc- 
tive shunts or leaks, and of practically cancelling it at a par- 
ticular frequency. How will this work out in the transmission 
of telephonic currents through the cable ? 

From the purely theoretical point of view one may be some- 
what prejudiced against the system at first. For, obviously, 
it is not a distortionless arrangement. That is got by balancing 
the self-induction of the main circuit against the lateral permit- 
tance, which can, subject to practical limitations, be done per- 
fectly. That is, there is a balance at any frequency, or the 
idea of periodic frequency does not enter at all. I doubt 
whether there can be any other distortionless circuit than that 
which (in the absence of another) I always refer to as the dis- 
tortionless circuit. With inductive shunts we produce a par- 
tial neutralisation of the permittance, the amount of which may 
vary very widely during the transmission of telephonic currents. 
Remember, too, that during the change from one frequency 
to another there are other phenomena than those concerned 
during the maintenance of a simply periodic state of variation. 

Nevertheless, we should be careful not to be prejudiced against 
an imperfect plan merely because it is so manifestly imperfect. 
It may perhaps be more easily ap d than a theoretically 
perfect system, and may possess practu d advantages of import- 
ance. Now this is a matter for practical experiment and expe- 
rience. But it is just here that there is an almost complete 
dearth of information. For although Pnff, Silvanus Thompson* 

has mentioned that he has made manv"experiments, yet he has 


* " Ocean Telephony," Chicago Congress, 1893. 


only described one case in which an inductive shunt was found 
good. It was an artificial cable of 7,000 ohms and 10 micro- 
farads. Presuming that there was a fairly good spreading of 
the permittance, we see that the permittance is that of about 
40 kilometres of a submarine cable, whilst the resistance is 
more like that of 1,000 or 2,000 kilometres. The inductance 
would probably be very small. We see that the example does 
not well represent a submarine cable. We should rather com- 
pare it with a very long overland circuit of very much smaller 
permittance per kilometre than the cable. Only, such a circuit, 
if properly put up, would have a large inductance, so that there 
is a failure here. However, it is mentioned that one shunt of 
312 ohms and a time-constant of 0'005 sec. rendered tele- 
phonic transmission possible except for shrill sounds. I cannot 
help thinking that Prof. Thompson has been much too reticent 
on the experimental side. He will perhaps furnish us with 
fuller information later on. At present there is little to 
judge by. 

General Theory of Transmission of Waves along a Circuit 
with or without Auxiliary Devices. 

221. In the meantime the following theory and formulae 
may perhaps be useful to those who may desire to go into the 
matter. I put the theory into such a form that it may be 
applied to various other cases of auxiliary arrangements. Let 
V and C be the transverse voltage and the current, at distance 
# at time t. Let their connections be 

where Z is a resistance operator, and Y a conductance operator, 
as described in 219, only now belonging to unit length of 
circuit. In the simplest p*-p Z reduces to a resistance, and Y 
to a conductance, per up length. But in general Z and Y 
may have various forms, in unlimited number, being then 
functions of electrical constants and of the time differentiator 
p. When there are no ai xiliary devices, and the natural resist- 
ance R, inductance L, leakance K, and permittance S (all per 
unit length) are constants, t .en we have 

.. (6) 



as before, 2014. But whatever the forms of Y and Z may 
be, in finite terms or transcendental, we may reduce them to the 
simple standard forms (6) when the simply periodic state of 
vibration is maintained, by employing the transformation 
p = ni, or p 2 = - n 2 . We see from this that we can practically 
examine cases of simple periodicity which might at first seem 
beyond all bounds of practicability. Note, however, that 
"forces" and "fluxes" should be in constant ratios, so that 
such phenomena as hysteresis, magnetic or electric, are ex- 

From equations (5) we see that the characteristic equation 
of Vis 

^ = YZV = ^V,say, .... (7) 
dx 2 

provided Y and Z are independent of x. This restriction is not 
a necessary one, but, of course, it makes an important practical 
simplification to have uniformity along the circuit. In (7) q 2 
represents the operator YZ, and in passing we may notice an 
interesting matter connected with partial differential equations 
of the type (7), or of the more general type 

V 2 V = 2 2 V, (8) 

appropriate to three dimensions in space. This type of equa- 
tion occurs in all sorts of physical problems, V being some 
variable which is propagated through a medium in a manner 
depending upon the nature of the operator q 2 , which involves 
the time-differentiator. It is usually a very simple rational 
function of p, such, for example, as to bring in only the first 
and second derivatives of V with respect to the time, and, so 
far as I know, no physical problems have presented themselves 
which involve an irrational partial differential equation for 
characteristic, as, for example, 

V 2 V = (a + 6p + cp 2 )V. .-. . . (9) 

It might, indeed, seem at first sight that a characteristic of 
this form was physically impossible,, being meaningless. 
Nevertheless, it is quite easy to see by the way equation (7) was 
constructed out of the components (5), that real physical 
problems may involve irrational characteristic equations, such 
as are exemplified by (9). For if either Y or Z be irrational, 


so is YZ. And we can easily (in imagination, not so easily in 
execution) choose auxiliary devices making Y or Z irrational. 
Similar remarks apply to (8). It results from the union of 
two distinct equations involving two variables, one of which is 
then eliminated to make the characteristic. If either of the 
component equations involves an irrational operator, the cha- 
racteristic resultant will be irrational. It must not be imagined 
that the solutions of such equations obtained by physical 
reasoning are impossible or beyond the range of mathematics, 
or that the results are physically meaningless. Some partial 
characteristics of the form (8), with irrational right members, 
I have examined and solved. I should imagine that there will 
probably be a field for equations of the kind in the future study 
of the complicated influence of matter upon phenomena which 
occur in their simplest manner in the ether away from matter, 
as in the theory of dispersion of light, for example, of electric 
absorption, and in similar subjects. 

Returning to equation (7), the general solution involves two 
arbitrary functions of the time, as in the form 

V = cosh<^.V + !^^ZC ..... (10) 

But it is preferable to take the case of disturbances propagated 
from a source into an infinitely long cable, in order to eliminate 
complications due to reflections and terminal apparatus. Then 

V = -*V ...... (11) 

is the solution representing V at x, t due to V given as a func- 
tion of the time at x = Q, the origin. And the current is 
given by 

by using the first of (5). 

To see the final state due to the continued action of constant 
V , give q, Y, and Z in (11), (12), the values they assume when 
p is put = in their general expressions, viz., for Y the steady 
leakance and for Z the steady resistance per unit length, and 
for q their geometric mean. To find how this state is arrived 
at, and, more generally, to interpret (11), (12) when V is any 
given function of the time, demands the conversion of these 



equations to proper algebraical form by execution of the analy- 
tical operations implied in q, Y, and Z. This we are not con- 
cerned with here, except as regards the case arising when V is 
a simply periodic function. Then put p = ni, and reduce Z to 
R + L#>, and Y to K + Sp, as before stated, equations (6), 
making (11), (12) become 

........ (13) 

where 2 = (R + I#)(K + Sp) ..... (15) 

These expressions have next to be reduced to the form 
(A + Bp)V , which is a matter of common algebra. When 
done we shall find that if 

P or Q = (J) [(R 2 + L 2 /* 2 )* (K 2 + S 2 ^ 2 )* (RK - LSrc 2 )]*, . (16) 
then the V solution is 

V = <J- p *sm(7i*-Qa>), ..... (17) 
due to V = e sin nt at the origin. From this, by the use of the 
first of (5), giving 

R-L dV 

we may derive the C solution. Or we may derive it by deve- 
loping (14). The result is 

C = j^ 8 [(KP + SnQ) sin + (SnP - KQ) coB](n - QOJ), (19) 
where the value of P 2 + Q 2 is given by 

. . (20) 

It may be only necessary to consider the amplitudes of V 
and C, and, of course, C is most important. That of V is 
obviously e e" Pa! , by (17). That of C may be got from (19), and 
is expressed by 

The wave speed is n/Q, the wave length 27T/Q, and the 
periodicity n/2ir. Another convenient form of (21) is 



where v = (LS)-*. 


Application of above Theory to Inductive Leakance. 

222. The above simply periodic solution is discussed in 
my "Electrical Papers," Vol. II. (especially p. 396 and p. 339), 
when R, L are the natural effective resistance and inductance, 
and K, S the leakance and permittance per unit length, the 
latter pair being supposed to be the same at all frequencies. 
(A paper by Prof. Perry, Phil. Mag., August, 1893, may also 
be consulted, but I do not think he is right in some of his 
conclusions.) But the same formulae are applicable when 
auxiliary devices are employed, if they are numerous enough, 
by a process analogous to the obvious one of representing 
large numbers of separate leaks by uniform leakance. 

Thus, in the case of inductive shunts, if we treat them as a 
set of separately located shunts, the full theory is very com- 
plicated, because it requires a separate formula for every section 
of the line. The best course to take is to distribute the 
inductive leakance uniformly, in imagination, of course. The 
result will represent the fullest possible carrying out of the 
principle concerned, and more. Thus, let /> be the resistance, 
and A, the inductance of the leak belonging to the unit length 
of cable. The reciprocal of p is of course the steady leakance 
per unit length. But it does not operate fully when the state 
is changing, owing to A, We shall now have 


where (/> + A-p)" 1 is the leakance operator, taking the place of 
p- 1 , whilst o- is the steady permittance per unit length. We 
have to use this special Y in the above formulae to represent 
the effect of inductive leakage. In the simply periodic case 
Y develops to 

The first term is the effective leakance, and the coefficient of p 
is the effective permittance per unit length. 

In the developed formulae (16), (17), (19) to (22), therefore, 
we must give K and S the values 

< 25 >